Values of L-Functions and Periods of Integrals

In this article, I present a conjecture (1.8, 2.8) relating the values of certain L-functions at certain integer points to the periods of integrals. The L-functions we consider here are those of motives a word to which we shall not attach a precise meaning. They include, notably, Artin L-functions, L-functions attached to algebraic Hecke characters (= Grossencharacter of type A0), and those attached to primitive holomorphic modular forms on the Poincare upper half plane (=new forms; we will consider all the L-functions, Lk, attached to the symmetric power, Sym , of the corresponding l-adic representation). This article owes its existence to D. Zagier : for his insistence on a conjecture, and for the experimental confirmation that he has provided for the L-functions L3 and L4 attached to ∆ = ∑ τ(n)q (see [18]). It is this confirmation that gave me the necessary confidence to verify that the conjecture is compatible with the results of Shimura [13] on the values of L-functions of algebraic Hecke characters.