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{{Short description|Subfield of number theory}}
{{DISPLAYTITLE:Special values of ''L''-functions}}
{{More citations needed|date=April 2019}}
In [[mathematics]], the study of '''special values of {{mvar|L}}-functions''' is a subfield of [[number theory]] devoted to generalising formulae such as the [[Leibniz formula for pi|Leibniz formula for {{pi}}]], namely
<math display=block>1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \frac{1}{9} \,-\, \cdots \;=\; \frac{\pi}{4},\!</math>

by the recognition that expression on the left-hand side is also <math>L(1)</math> where <math>L(s)</math> is the [[Dirichlet L-function|Dirichlet {{mvar|L}}-function]] for the field of [[Gaussian rational]] numbers. This formula is a special case of the [[analytic class number formula]], and in those terms reads that the Gaussian field has [[class number problem|class number 1]]. The factor <math>\tfrac14</math> on the right hand side of the formula corresponds to the fact that this field contains four [[roots of unity]].

==Conjectures==
There are two families of conjectures, formulated for general classes of [[L-function|{{mvar|L}}-functions]] (the very general setting being for {{mvar|L}}-functions associated to [[Chow motive]]s over [[number field]]s), the division into two reflecting the questions of:
<ol type="a">
<li> how to replace <math>\pi</math> in the [[Leibniz formula for pi|Leibniz formula]] by some other "transcendental" number (regardless of whether it is currently possible for [[transcendental number theory]] to provide a proof of the transcendence); and</li>
<li> how to generalise the rational factor in the formula (class number divided by number of roots of unity) by some algebraic construction of a rational number that will represent the ratio of the {{mvar|L}}-function value to the "transcendental" factor.</li>
</ol>
Subsidiary explanations are given for the integer values of <math>n</math> for which a formulae of this sort involving <math>L(n)</math> can be expected to hold.

The conjectures for (a) are called ''Beilinson's conjectures'', for [[Alexander Beilinson]].<ref>[http://wwwmath.uni-muenster.de/u/pschnei/publ/beilinson-volume/Schneider.pdf Peter Schneider, ''Introduction to the Beilinson Conjectures'' (PDF)]</ref><ref>[http://webusers.imj-prg.fr/~jan.nekovar/pu/mot.pdf Jan Nekovář, ''Beilinson's Conjectures'' (PDF)]</ref> The idea is to abstract from the [[regulator of a number field]] to some "higher regulator" (the [[Beilinson regulator]]), a determinant constructed on a real vector space that comes from [[algebraic K-theory]].

The conjectures for (b) are called the ''Bloch–Kato conjectures for special values ''(for [[Spencer Bloch]] and [[Kazuya Kato]]; this circle of ideas is distinct from the [[Bloch–Kato conjecture]] of K-theory, extending the [[Milnor conjecture (K-theory)|Milnor conjecture]], a proof of which was announced in 2009). They are also called the ''Tamagawa number conjecture'', a name arising via the [[Birch–Swinnerton-Dyer conjecture]] and its formulation as an [[elliptic curve]] analogue of the [[Tamagawa number]] problem for [[linear algebraic group]]s.<ref>[http://www.dpmms.cam.ac.uk/Seminars/Kuwait/abstracts/L56.pdf Matthias Flach, ''The Tamagawa Number Conjecture'' (PDF)]</ref> In a further extension, the equivariant Tamagawa number conjecture (ETNC) has been formulated, to consolidate the connection of these ideas with [[Iwasawa theory]], and its so-called [[Main_conjecture_of_Iwasawa_theory|Main Conjecture]].

===Current status===
All of these conjectures are known to be true only in special cases.

==See also==
*[[Brumer–Stark conjecture]]

==Notes==
{{reflist}}

==References==
* {{Citation | last1=Kings | first1=Guido | title=The Bloch–Kato conjecture on special values of ''L''-functions. A survey of known results | url=http://jtnb.cedram.org/item?id=JTNB_2003__15_1_179_0 |mr=2019010 | year=2003 | journal=Journal de théorie des nombres de Bordeaux | issn=1246-7405 | volume=15 | issue=1 | pages=179–198 | doi= 10.5802/jtnb.396| doi-access=free }}
*{{Springer|id= b/b110220|title = Beilinson conjectures}}
*{{Springer|id= K/k055000|title = K-functor in algebraic geometry}}
*{{cite arXiv
|first1=Richard J.
|last1=Mathar
|title=Table of Dirichlet L-Series and Prime Zeta Modulo Functions for small moduli
|eprint=1008.2547
|year=2010
<!--|bibcode=2010arXiv1008.2547M-->
|class=math.NT
|mode=cs2
}}

==External links==
* [http://www.mpg.de/459553/L-Funktionen_gesamt.pdf L-funktionen und die Vermutingen von Deligne und Beilinson (L-functions and the conjectures of Deligne and Beilsnson)]

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[[Category:Zeta and L-functions]]