Values of $L$-functions at the critical point

Critical Values of Symmetric Power L-functions

We consider the critical values of symmetric power L-functions attached to elliptic curves over Q. We show how to calculate a canonical Deligne period, and in several numerical examples, especially for sixth and tenth powers, we examine the factorisation of the rational number apparently obtained when one divides the critical value by the canonical period. This seems to provide some support for...

Distribution of Values of L-functions at the Edge of the Critical Strip

Abstract. We prove several results on the distribution of values of L-functions at the edge of the critical strip, by constructing and studying a large class of random Euler products. Among new applications, we provide a precise estimate for the distribution of the values at s = 1 of the family of k-th symmetric power L-functions of primitive cusp forms of weight 2 in the level aspect (assuming...

On Critical Values of L-functions of Potentially Automorphic Motives

In this paper we prove a version of Deligne’s conjecture for potentially automorphic motives, twisted by certain algebraic Hecke characters. The Hecke characters are chosen in such a way that we can use automorphic methods in the context of totally definite unitary groups.

General Selmer Groups and Critical Values of Hecke L-functions

Let K be an imaginary quadratic eld and let O be the ring of integers of K. Let E be an elliptic curve deened over Q with complex multiplication by O. Let be the Grr ossencharacter attached to the curve E over K by the theory of complex multiplication, and let L(k ; s) be the complex Hecke L-function attached to the powers of , k = 1; 2; ; here we have xed an embedding of K in C. In a previous ...

Non-vanishing of Dirichlet L-functions at the Central Point