View source for Particular values of the Riemann zeta function

{{Short description|Constants of the mathematical zeta function}}
{{Use American English|date = March 2019}}
In [[mathematics]], the [[Riemann zeta function]] is a function in [[complex analysis]], which is also important in [[number theory]]. It is often denoted <math>\zeta(s)</math> and is named after the mathematician [[Bernhard Riemann]]. When the argument <math>s</math> is a [[real number]] greater than one, the zeta function satisfies the equation
<math display="block">\zeta(s) = \sum_{n=1}^\infty\frac{1}{n^s} \, .</math>
It can therefore provide the sum of various convergent [[Series (mathematics)|infinite series]], such as <math display="inline">\zeta(2) = \frac{1}{1^2} + </math><math display="inline">\frac{1}{2^2} + </math><math display="inline">\frac{1}{3^2} + \ldots \, .</math> Explicit or numerically efficient formulae exist for <math>\zeta(s)</math> at integer arguments, all of which have real values, including this example. This article lists these formulae, together with tables of values. It also includes derivatives and some series composed of the zeta function at integer arguments.

The same equation in <math>s</math> above also holds when <math>s</math> is a [[complex number]] whose [[real part]] is greater than one, ensuring that the infinite sum still converges. The zeta function can then be extended to the whole of the [[complex plane]] by [[analytic continuation]], except for a [[simple pole]] at <math>s=1</math>. The [[complex derivative]] exists in this more general region, making the zeta function a [[meromorphic function]]. The above equation no longer applies for these extended values of <math>s</math>, for which the corresponding summation would diverge. For example, the full zeta function exists at <math>s=-1</math> (and is therefore finite there), but the corresponding series would be <math display="inline">1 + 2 + 3 + \ldots \, ,</math> whose [[partial sum]]s would grow indefinitely large.

The zeta function values listed below include function values at the negative even numbers ({{math|''s'' {{=}} &minus;2}}, {{nowrap|{{math|&minus;4}}, etc.}}), for which {{math|''&zeta;''(''s'') {{=}} 0}} and which make up the so-called '''trivial zeros'''. The '''[[Riemann zeta function]]''' article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane. The successful characterisation of its non-trivial zeros in the wider plane is important in number theory, because of the [[Riemann hypothesis]].

==The Riemann zeta function at 0 and 1==
At [[Zero (complex analysis)|zero]], one has
<math display="block">\zeta(0)= {B^-_1}=-{B^+_1}=-\tfrac{1}{2}\!</math>

At 1 there is a [[Pole (complex analysis)|pole]], so ''ζ''(1) is not finite but the left and right limits are:
<math display="block">\lim_{\varepsilon\to 0^\pm}\zeta(1+\varepsilon) = \pm\infty</math>
Since it is a pole of first order, it has a [[Residue (complex analysis)|complex residue]]
<math display="block">\lim_{\varepsilon\to 0} \varepsilon \zeta(1+\varepsilon) = 1\,.</math>

==Positive integers==

===Even positive integers===
For the even positive integers <math>n</math>, one has the relationship to the [[Bernoulli numbers]] <math>B_n</math>:

<math display="block">\zeta(n) = (-1)^{\tfrac{n}{2}+1}\frac{(2\pi)^{n}B_n}{2(n!)} \,.</math>

The computation of <math>\zeta(2)</math> is known as the [[Basel problem]]. The value of <math>\zeta(4)</math> is related to the [[Stefan–Boltzmann law]] and [[Wien approximation]] in physics. The first few values are given by:
<math display="block">
\begin{align}
\zeta(2) & = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6} \\[4pt]
\zeta(4) & = 1 + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90} \\[4pt]
\zeta(6) & = 1 + \frac{1}{2^6} + \frac{1}{3^6} + \cdots = \frac{\pi^6}{945} \\[4pt]
\zeta(8) & = 1 + \frac{1}{2^8} + \frac{1}{3^8} + \cdots = \frac{\pi^8}{9450} \\[4pt]
\zeta(10) & = 1 + \frac{1}{2^{10}} + \frac{1}{3^{10}} + \cdots = \frac{\pi^{10}}{93555} \\[4pt]
\zeta(12) & = 1 + \frac{1}{2^{12}} + \frac{1}{3^{12}} + \cdots = \frac{691\pi^{12}}{638512875} \\[4pt]
\zeta(14) & = 1 + \frac{1}{2^{14}} + \frac{1}{3^{14}} + \cdots = \frac{2\pi^{14}}{18243225} \\[4pt]
\zeta(16) & = 1 + \frac{1}{2^{16}} + \frac{1}{3^{16}} + \cdots = \frac{3617\pi^{16}}{325641566250}\,.
\end{align}
</math>

Taking the limit <math>n \rightarrow \infty</math>, one obtains <math>\zeta (\infty) = 1</math>.

{| class="wikitable"
|+ Selected values for even integers
|-
! scope="col" | Value
! scope="col" | Decimal expansion
! scope="col" | Source
|-
| <math>\zeta(2)</math>
| {{val|1.6449340668482264364}}... 
| {{OEIS2C|A013661}}
|-
| <math>\zeta(4)</math>
| {{val|1.0823232337111381915}}...
| {{OEIS2C|A013662}}
|-
| <math>\zeta(6)</math>
| {{val|1.0173430619844491397}}...
| {{OEIS2C|A013664}}
|-
| <math>\zeta(8)</math>
| {{val|1.0040773561979443393}}...
| {{OEIS2C|A013666}}
|-
| <math>\zeta(10)</math>
| {{val|1.0009945751278180853}}...
| {{OEIS2C|A013668}}
|-
| <math>\zeta(12)</math>
| {{val|1.0002460865533080482}}...
| {{OEIS2C|A013670}}
|-
| <math>\zeta(14)</math>
| {{val|1.0000612481350587048}}...
| {{OEIS2C|A013672}}
|-
| <math>\zeta(16)</math>
| {{val|1.0000152822594086518}}...
| {{OEIS2C|A013674}}
|}

The relationship between zeta at the positive even integers and powers of pi may be written as

<math display="block">a_n \zeta(2n) = \pi^{2n} b_n</math>

where <math>a_n</math> and <math>b_n</math> are coprime positive integers for all <math>n</math>. These are given by the integer sequences {{OEIS2C|id=A002432}} and {{OEIS2C|id=A046988}}, respectively, in [[OEIS]].  Some of these values are reproduced below:

{| class="wikitable"
|+ coefficients
|-
! ''n''
! ''a''<sub>''n''</sub>
! ''b''<sub>''n''</sub>
|-
| 1
| 6
| 1
|-
| 2
| 90
| 1
|-
| 3
| 945
| 1
|-
| 4
| 9450
| 1
|-
| 5
| 93555
| 1
|-
| 6
| 638512875
| 691
|-
| 7
| 18243225
| 2
|-
| 8
| 325641566250
| 3617
|-
| 9
| 38979295480125
| 43867
|-
| 10
| 1531329465290625
| 174611
|-
| 11
| 13447856940643125
| 155366
|-
| 12
| 201919571963756521875
| 236364091
|-
| 13
| 11094481976030578125
| 1315862
|-
| 14
| 564653660170076273671875
| 6785560294
|-
| 15
| 5660878804669082674070015625
| 6892673020804
|-
| 16
| 62490220571022341207266406250
| 7709321041217
|-
| 17
| 12130454581433748587292890625
| 151628697551
|}

If we let <math>\eta_n=b_n/a_n</math> be the coefficient of <math>\pi^{2n}</math> as above,
<math display="block">\zeta(2n) = \sum_{\ell=1}^{\infty}\frac{1}{\ell^{2n}}=\eta_n\pi^{2n}</math>
then we find recursively,

<math display="block">
\begin{align}
\eta_1 &= 1/6 \\
\eta_n &= \sum_{\ell=1}^{n-1}(-1)^{\ell-1}\frac{\eta_{n-\ell}}{(2\ell+1)!}+(-1)^{n+1}\frac{n}{(2n+1)!}
\end{align}
</math>

This recurrence relation may be derived from that for the [[Bernoulli number]]s.

Also, there is another recurrence:

<math display="block">\zeta(2n)=\frac{1}{n+\frac{1}{2}} \sum_{k=1}^{n-1} \zeta(2k)\zeta(2n-2k) \quad \text{ for } \quad n>1 </math>
which can be proved, using that <math>\frac{d}{dx} \cot(x) = -1-\cot^{2} (x) </math>

The values of the zeta function at non-negative even integers have the [[generating function]]:
<math display="block">\sum_{n=0}^\infty \zeta(2n) x^{2n} = -\frac{\pi x}{2} \cot(\pi x) = -\frac{1}{2} + \frac{\pi^2}{6} x^2 + \frac{\pi^4}{90} x^4+\frac{\pi^6}{945}x^6 + \cdots</math>
Since 
<math display="block">\lim_{n\rightarrow\infty} \zeta(2n)=1</math> 
The formula also shows that for  <math> n\in\mathbb{N}, n\rightarrow\infty</math>,
<math display="block">\left|B_{2n}\right| \sim \frac{(2n)!\,2}{\;~(2\pi)^{2n}\,}</math>

===Odd positive integers===
The sum of the [[harmonic series (mathematics)|harmonic series]] is infinite.
<math = display="block">\zeta(1) = 1 + \frac{1}{2} + \frac{1}{3} + \cdots = \infty\!</math>

The value {{math|''ζ''(3)}} is also known as [[Apéry's constant]] and has a role in the electron's gyromagnetic ratio.
The value {{math|''ζ''(3)}} also appears in [[Planck's law#Properties|Planck's law]].
These and additional values are:
{| class="wikitable"
|+ Selected values for odd integers
|-
! scope="col" | Value
! scope="col" | Decimal expansion
! scope="col" | Source
|-
| [[Apéry's constant|<math>\zeta(3)</math>]]
| {{val|1.2020569031595942853}}...
| {{OEIS2C|A02117}}
|-
| <math>\zeta(5)</math>
| {{val|1.0369277551433699263}}...
| {{OEIS2C|A013663}}
|-
| <math>\zeta(7)</math>
| {{val|1.0083492773819228268}}...
| {{OEIS2C|A013665}}
|-
| <math>\zeta(9)</math>
| {{val|1.0020083928260822144}}...
| {{OEIS2C|A013667}}
|-
| <math>\zeta(11)</math>
| {{val|1.0004941886041194645}}...
| {{OEIS2C|A013669}}
|-
| <math>\zeta(13)</math>
| {{val|1.0001227133475784891}}...
| {{OEIS2C|A013671}}
|-
| <math>\zeta(15)</math>
| {{val|1.0000305882363070204}}...
| {{OEIS2C|A013673}}
|}

It is known that {{math|''ζ''(3)}} is irrational ([[Apéry's theorem]]) and that infinitely many of the numbers {{math|''ζ''(2''n''&nbsp;+&nbsp;1) : ''n'' ∈ <math>\mathbb{N}</math> }}, are irrational.<ref>{{cite journal |last=Rivoal |first=T. |authorlink=Tanguy Rivoal|year=2000 |title=La fonction zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs |journal=Comptes Rendus de l'Académie des Sciences, Série I |volume=331 |issue=4 |pages=267–270 |bibcode=2000CRASM.331..267R |doi=10.1016/S0764-4442(00)01624-4 |arxiv=math/0008051|s2cid=119678120 }}</ref> There are also results on the irrationality of values of the Riemann zeta function at the elements of certain subsets of the positive odd integers; for example, at least one of {{math|''ζ''(5), ''ζ''(7), ''ζ''(9), or ''ζ''(11)}} is irrational.<ref>{{cite journal |author=W. Zudilin |title=One of the numbers ''ζ''(5), ''ζ''(7), ''ζ''(9), ''ζ''(11) is irrational |journal=Russ. Math. Surv. |year=2001 |volume=56 |issue=4 |pages=774–776 |doi=10.1070/rm2001v056n04abeh000427|bibcode=2001RuMaS..56..774Z |s2cid=250734661 }}</ref>

The positive odd integers of the zeta function appear in physics, specifically [[Correlation function (statistical mechanics)|correlation functions]] of antiferromagnetic [[Heisenberg model (quantum)|XXX spin chain]].<ref>{{cite journal |first1=H.E. |last1=Boos |first2=V.E. |last2=Korepin |first3=Y. |last3=Nishiyama |first4=M. |last4=Shiroishi |year=2002 |title=Quantum correlations and number theory |journal=J. Phys. A |volume=35 |issue=20 |pages=4443–4452 |arxiv=cond-mat/0202346 |bibcode=2002JPhA...35.4443B |doi=10.1088/0305-4470/35/20/305|s2cid=119143600 }}.</ref>

Most of the identities following below are provided by [[Simon Plouffe]]. They are notable in that they converge quite rapidly, giving almost three digits of precision per iteration, and are thus useful for high-precision calculations.

Plouffe stated the following identities without proof.<ref>{{cite web | url=http://www.plouffe.fr/simon/identities.html | title=Identities for Zeta(2*n+1) }}</ref> Proofs were later given by other authors.<ref>{{cite web | url=https://cs.uwaterloo.ca/journals/JIS/VOL14/Lopatto/lopatto2.html | title=Formulas for Odd Zeta Values and Powers of Pi }}</ref>

====''&zeta;''(5)====

<math display="block">
\begin{align}
\zeta(5)&=\frac{1}{294}\pi^5 -\frac{72}{35} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} -1)}-\frac{2}{35} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} +1)}\\
\zeta(5)&=12 \sum_{n=1}^\infty \frac{1}{n^5 \sinh (\pi n)} -\frac{39}{20} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} -1)}+\frac{1}{20} \sum_{n=1}^\infty \frac{1}{n^5 (e^{2\pi n} +1)}
\end{align}
</math>

====''&zeta;''(7)====
<math display="block">\zeta(7)=\frac{19}{56700}\pi^7 - 2 \sum_{n=1}^\infty \frac{1}{n^7 (e^{2\pi n} -1)}\!</math>

Note that the sum is in the form of a [[Lambert series]].

====''&zeta;''(2''n'' + 1)====
By defining the quantities

<math display="block">S_\pm(s) = \sum_{n=1}^\infty \frac{1}{n^s (e^{2\pi n} \pm 1)}</math>

a series of relationships can be given in the form

<math display="block">0=a_n \zeta(n) - b_n \pi^{n} + c_n S_-(n) + d_n S_+(n)</math>

where ''a''<sub>''n''</sub>, ''b''<sub>''n''</sub>, ''c''<sub>''n''</sub> and ''d''<sub>''n''</sub> are positive integers. Plouffe gives a table of values:

{| class="wikitable"
|+ coefficients
|-
! ''n''
! ''a''<sub>''n''</sub>
! ''b''<sub>''n''</sub>
! ''c''<sub>''n''</sub>
! ''d''<sub>''n''</sub>
|-
| 3
| 180
| 7
| 360
| 0
|-
| 5
| 1470
| 5
| 3024
| 84
|-
| 7
| 56700
| 19
| 113400
| 0
|-
| 9
| 18523890
| 625
| 37122624
| 74844
|-
| 11
| 425675250
| 1453
| 851350500
| 0
|-
| 13
| 257432175
| 89
| 514926720
| 62370
|-
| 15
| 390769879500
| 13687
| 781539759000
| 0
|-
| 17
| 1904417007743250
| 6758333
| 3808863131673600
| 29116187100
|-
| 19
| 21438612514068750
| 7708537
| 42877225028137500
| 0
|-
| 21
| 1881063815762259253125
| 68529640373
| 3762129424572110592000
| 1793047592085750
|}

These integer constants may be expressed as sums over Bernoulli numbers, as given in (Vepstas, 2006) below.
 
A fast algorithm for the calculation of Riemann's zeta function for any integer argument is given by E. A. Karatsuba.<ref>{{cite journal
|first1=E. A.
|last1=Karatsuba
|title=Fast calculation of the Riemann zeta function ''ζ''(''s'') for integer values of the argument&nbsp;''s''
|year=1995
|journal=Probl. Perdachi Inf.
|volume=31
|issue=4
|pages=69–80
|mr=1367927
|url=http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=ppi&paperid=294&option_lang=eng
}}</ref><ref>E. A. Karatsuba: Fast computation of the Riemann zeta function for integer argument. Dokl. Math. Vol.54, No.1, p. 626 (1996).</ref><ref>E. A. Karatsuba: Fast evaluation of ''ζ''(3).  Probl. Inf. Transm. Vol.29, No.1, pp. 58–62 (1993).</ref>

==Negative integers==
In general, for negative integers (and also zero), one has

<math display="block">\zeta(-n)=(-1)^{n}\frac{B_{n+1}}{n+1}</math>

The so-called "trivial zeros" occur at the negative even integers:

<math display="block">\zeta(-2n)=0</math> ([[Ramanujan summation]])

The first few values for negative odd integers are

<math display="block">
\begin{align}
\zeta(-1) &=-\frac{1}{12} \\[4pt]
\zeta(-3) &=\frac{1}{120} \\[4pt]
\zeta(-5) &=-\frac{1}{252} \\[4pt]
\zeta(-7) &=\frac{1}{240} \\[4pt]
\zeta(-9) &= -\frac{1}{132} \\[4pt]
\zeta(-11)&= \frac{691}{32760} \\[4pt]
\zeta(-13)&= -\frac{1}{12} 
\end{align}
</math>

However, just like the [[Bernoulli numbers]], these do not stay small for increasingly negative odd values. For details on the first value, see [[1 + 2 + 3 + 4 + · · ·]].

So ''ζ''(''m'') can be used as the definition of all (including those for index 0 and 1) Bernoulli numbers.

==Derivatives==
The derivative of the zeta function at the negative even integers is given by

<math display="block">\zeta^{\prime}(-2n) = (-1)^n \frac {(2n)!} {2 (2\pi)^{2n}} \zeta (2n+1)\,.</math>

The first few values of which are

<math display="block">
\begin{align}
\zeta^{\prime}(-2) & = -\frac{\zeta(3)}{4\pi^2} \\[4pt]
\zeta^{\prime}(-4) & = \frac{3}{4\pi^4} \zeta(5) \\[4pt]
\zeta^{\prime}(-6) & = -\frac{45}{8\pi^6} \zeta(7) \\[4pt]
\zeta^{\prime}(-8) & = \frac{315}{4\pi^8} \zeta(9)\,.
\end{align}
</math>

One also has

<math display="block">
\begin{align}
\zeta^{\prime}(0) & = -\frac{1}{2}\ln(2\pi) \\[4pt]
\zeta^{\prime}(-1) & = \frac{1}{12}-\ln A \\[4pt]
\zeta^{\prime}(2) & = \frac{1}{6}\pi^2(\gamma +\ln 2-12\ln A+\ln \pi)
\end{align}
</math>

where ''A'' is the [[Glaisher–Kinkelin constant]].  The first of these identities implies that the regularized product of the reciprocals of the positive integers is <math>1/\sqrt{2\pi}</math>, thus the amusing "equation" <math>\infty!=\sqrt{2\pi}</math>.<ref>{{citation|author1=Muñoz García, E.|author2=Pérez Marco, R.|title=The Product Over All Primes is <math>4\pi^2</math>|journal=Commun. Math. Phys.|issue=277|pages=69–81|year=2008}}.</ref>

From the logarithmic derivative of the functional equation,

<math display="block">2\frac{\zeta'(1/2)}{\zeta(1/2)} = \log(2\pi)+\frac{\pi\cos(\pi/4)}{2\sin(\pi/4)}-\frac{\Gamma'(1/2)}{\Gamma(1/2)}=\log(2\pi)+\frac{\pi}{2}+2\log 2+\gamma\,.</math>

{| class="wikitable"
|+ Selected derivatives
|-
! scope="col" | Value
! scope="col" | Decimal expansion
! scope="col" | Source
|-
| <math>\zeta'(3)</math>
| {{val|-0.19812624288563685333}}...
| {{OEIS2C|A244115}}
|-
| <math>\zeta'(2)</math>
| {{val|-0.93754825431584375370}}...
| {{OEIS2C|A073002}}
|-
| <math>\zeta'(0)</math>
| {{val|-0.91893853320467274178}}...
| {{OEIS2C|A075700}}
|-
| <math>\zeta'(-\tfrac{1}{2})</math>
| {{val|-0.36085433959994760734}}...
| {{OEIS2C|A271854}}
|-
| <math>\zeta'(-1)</math>
| {{val|-0.16542114370045092921}}...
| {{OEIS2C|A084448}}
|-
| <math>\zeta'(-2)</math>
| {{val|-0.030448457058393270780}}...
| {{OEIS2C|A240966}}
|-
| <math>\zeta'(-3)</math>
| {{val|+0.0053785763577743011444}}...
| {{OEIS2C|A259068}}
|-
| <math>\zeta'(-4)</math>
| {{val|+0.0079838114502686242806}}...
| {{OEIS2C|A259069}}
|-
| <math>\zeta'(-5)</math>
| {{val|-0.00057298598019863520499}}...
| {{OEIS2C|A259070}}
|-
| <math>\zeta'(-6)</math>
| {{val|-0.0058997591435159374506}}...
| {{OEIS2C|A259071}}
|-
| <math>\zeta'(-7)</math>
| {{val|-0.00072864268015924065246}}...
| {{OEIS2C|A259072}}
|-
| <math>\zeta'(-8)</math>
| {{val|+0.0083161619856022473595}}...
| {{OEIS2C|A259073}}
|}

==Series involving ''&zeta;''(''n'')==
The following sums can be derived from the generating function:
<math display="block">\sum_{k=2}^\infty \zeta(k) x^{k-1}=-\psi_0(1-x)-\gamma</math>
where {{math|''ψ''<sub>0</sub>}} is the [[digamma function]].

<math display="block">
\begin{align}
\sum_{k=2}^\infty (\zeta(k) -1) & = 1 \\[4pt]
\sum_{k=1}^\infty (\zeta(2k) -1) & = \frac{3}{4} \\[4pt]
\sum_{k=1}^\infty (\zeta(2k+1) -1) & = \frac{1}{4} \\[4pt]
\sum_{k=2}^\infty (-1)^k(\zeta(k) -1) & = \frac{1}{2}
\end{align}
</math>

Series related to the [[Euler–Mascheroni constant]] (denoted by {{math|''γ''}}) are
<math display="block">
\begin{align}
\sum_{k=2}^\infty (-1)^k \frac{\zeta(k)}{k} & = \gamma \\[4pt]
\sum_{k=2}^\infty \frac{\zeta(k) - 1}{k} & = 1 - \gamma \\[4pt]
\sum_{k=2}^\infty (-1)^k \frac{\zeta(k)-1}{k} & = \ln2 + \gamma - 1
\end{align}
</math>

and using the principal value 
<math display="block">\zeta(k) = \lim_{\varepsilon \to 0} \frac{\zeta(k+\varepsilon)+\zeta(k-\varepsilon)}{2}</math>
which of course affects only the value at 1, these formulae can be stated as

<math display="block">
\begin{align}
\sum_{k=1}^\infty (-1)^k \frac{\zeta(k)}{k} & = 0 \\[4pt]
\sum_{k=1}^\infty \frac{\zeta(k) - 1}{k} & = 0 \\[4pt]
\sum_{k=1}^\infty (-1)^k \frac{\zeta(k)-1}{k} & = \ln2
\end{align}
</math>

and show that they depend on the principal value of {{nowrap|{{math|''ζ''(1) {{=}} ''γ''}} .}}

== Nontrivial zeros ==
{{main|Riemann hypothesis}}

Zeros of the Riemann zeta except negative even integers are called "nontrivial zeros". The [[Riemann hypothesis]] states that the real part of every nontrivial zero must be {{sfrac|1|2}}. In other words, all known nontrivial zeros of the Riemann zeta are of the form {{math|1=''z'' = {{sfrac|1|2}} + ''y''i}} where ''y'' is a real number. The following table contains the decimal expansion of Im(''z'') for the first few nontrivial zeros:

{| class="wikitable"
|+ Selected nontrivial zeros
|-
! scope="col" | Decimal expansion of Im(''z'')
! scope="col" | Source
|-
| {{val|14.134725141734693790}}... 
| {{OEIS2C|A058303}}
|-
| {{val|21.022039638771554992}}...
| {{OEIS2C|A065434}}
|-
| {{val|25.010857580145688763}}...
| {{OEIS2C|A065452}}
|-
| {{val|30.424876125859513210}}...
| {{OEIS2C|A065453}}
|-
| {{val|32.935061587739189690}}...
| {{OEIS2C|A192492}}
|-
| {{val|37.586178158825671257}}...
| {{OEIS2C|A305741}}
|-
| {{val|40.918719012147495187}}...
| {{OEIS2C|A305742}}
|-
| {{val|43.327073280914999519}}...
| {{OEIS2C|A305743}}
|-
| {{val|48.005150881167159727}}...
| {{OEIS2C|A305744}}
|-
| {{val|49.773832477672302181}}...
| {{OEIS2C|A306004}}
|}

[[Andrew Odlyzko]] computed the first 2 million nontrivial zeros accurate to within 4{{x10^|-9}}, and the first 100 zeros accurate within 1000 decimal places. See their website for the tables and bibliographies.<ref>{{cite web |last1=Odlyzko |first1=Andrew |title=Tables of zeros of the Riemann zeta function |url=http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html |access-date=7 September 2022}}</ref><ref>{{cite web |last1=Odlyzko |first1=Andrew |title=Papers on Zeros of the Riemann Zeta Function and Related Topics |url=http://www.dtc.umn.edu/~odlyzko/doc/zeta.html |access-date=7 September 2022}}</ref>
A table of about 103 billion zeros with high precision (of ±2<sup>−102</sup>≈±2·10<sup>−31</sup>) is available for interactive access and download (although in a very inconvenient compressed format) via '''LMFDB'''.<ref>[https://www.lmfdb.org/zeros/zeta/ LMFDB: Zeros of ζ(''s'')]</ref>

== Ratios ==
Although evaluating particular values of the zeta function is difficult, often certain ratios can be found by inserting [[particular values of the gamma function]] into the functional equation

<math>
\zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)
</math>

We have simple relations for half-integer arguments

<math>
\begin{align}
\frac{\zeta(3/2)}{\zeta(-1/2)} &= -4\pi \\
\frac{\zeta(5/2)}{\zeta(-3/2)} &= -\frac{16\pi^2}{3} \\
\frac{\zeta(7/2)}{\zeta(-5/2)} &= \frac{64\pi^3}{15} \\
\frac{\zeta(9/2)}{\zeta(-7/2)} &= \frac{256\pi^4}{105}
\end{align}
</math>

Other examples follow for more complicated evaluations and relations of the gamma function.  For example a consequence of the relation

<math>\Gamma\left(\tfrac{3}{4}\right) =\left(\tfrac{\pi} {2}\right) ^{\tfrac{1}{4}} {\operatorname{AGM}\left(\sqrt 2, 1\right)}^{\tfrac{1}{2}}</math>

is the zeta ratio relation

<math>
\frac{\zeta(3/4)}{\zeta(1/4)} = 2\sqrt{\frac{\pi}{(2-\sqrt{2})\operatorname{AGM}\left(\sqrt 2, 1\right)}}
</math>

where AGM is the [[arithmetic–geometric mean]].  In a similar vein, it is possible to form radical relations, such as from

:<math>\frac{\Gamma\left(\frac{1}{5}\right)^2}{\Gamma\left(\frac{1}{10}\right)\Gamma\left(\frac{3}{10}\right)} = \frac{\sqrt{1+\sqrt{5}}}{2^{\tfrac{7}{10}}\sqrt[4]{5}}</math>

the analogous zeta relation is

<math>
\frac{\zeta(1/5)^2\zeta(7/10)\zeta(9/10)}{\zeta(1/10)\zeta(3/10)\zeta(4/5)^2} = \frac{(5-\sqrt{5})\left(\sqrt{10}+\sqrt{5+\sqrt{5}}\right)}{10\cdot2^{\tfrac{3}{10}}}
</math>

==References==
{{reflist|25em}}

==Further reading==
* {{cite journal|last1=Ciaurri|first1=Óscar|last2=Navas|first2=Luis M.|last3=Ruiz|first3=Francisco J.|last4=Varona|first4=Juan L.|title=A Simple Computation of ''ζ''(2''k'')|journal=The American Mathematical Monthly|date=May 2015|volume=122|issue=5|pages=444–451|jstor=10.4169/amer.math.monthly.122.5.444|doi=10.4169/amer.math.monthly.122.5.444|arxiv=1209.5030 |s2cid=207521195}}
* [[Simon Plouffe]], "[http://www.lacim.uqam.ca/~plouffe/identities.html Identities inspired from Ramanujan Notebooks] {{Webarchive|url=https://web.archive.org/web/20090130142844/http://www.lacim.uqam.ca/~plouffe/identities.html |date=2009-01-30 }}", (1998).
* [[Simon Plouffe]], "[http://www.lacim.uqam.ca/~plouffe/inspired22.html Identities inspired by Ramanujan Notebooks part 2] [http://www.lacim.uqam.ca/~plouffe/inspired2.pdf PDF] {{Webarchive|url=https://web.archive.org/web/20110926231624/http://www.lacim.uqam.ca/~plouffe/inspired2.pdf |date=2011-09-26 }}" (2006).
* {{cite journal
|first1=Linas
|last1=Vepstas
|url=http://www.linas.org/math/plouffe-ram.pdf
|title=On Plouffe's Ramanujan identities
|journal=The Ramanujan Journal
|arxiv=math.NT/0609775
|year=2006
|volume=27
|issue=3
|pages=387–408
|doi=10.1007/s11139-011-9335-9
|s2cid=8789411
}}
* {{cite journal
|first1=Wadim
|last1=Zudilin
|authorlink=Wadim Zudilin
|title=One of the Numbers ''ζ''(5), ''ζ''(7), ''ζ''(9), ''ζ''(11) Is Irrational
|journal=[[Russian Mathematical Surveys]]
|volume= 56
|pages=774&ndash;776
|year=2001
|issue=4
|doi=10.1070/RM2001v056n04ABEH000427
|mr=1861452
|bibcode=2001RuMaS..56..774Z
|s2cid=250734661
}} [http://wain.mi-ras.ru/PS/zeta5-11$.pdf PDF] [http://wain.mi-ras.ru/PS/zeta5-11.pdf PDF Russian] [http://wain.mi-ras.ru/PS/zeta5-11.ps.gz PS Russian]
* Nontrival zeros reference by [[Andrew Odlyzko]]:
** [http://www.dtc.umn.edu/~odlyzko/doc/zeta.html Bibliography]
** [http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html Tables]

[[Category:Mathematical constants]]
[[Category:Zeta and L-functions]]
[[Category:Irrational numbers]]