In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for π, namely
is the Dirichlet 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 1.
on the right hand side of the formula corresponds to the fact that this field contains four roots of unity.
There are two families of conjectures, formulated for general classes of L-functions (the very general setting being for L-functions associated to Chow motives over number fields), the division into two reflecting the questions of: Subsidiary explanations are given for the integer values of
[1][2] 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.
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 groups.
[3] 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.