Certain L-functions at s = 1/2

NON - VANISHING OF QUADRATIC DIRICHLET L - FUNCTIONS AT s = 12 3

Nonvanishing of certain Rankin-Selberg L-functions

In this article we prove that given a holomorphic cusp form f and any point s0 in the complex plane, there is a holomorphic cusp form g such that the Rankin-Selberg L-function L(s, f × g) is non-zero at s0. Résumé: Dans cet article, on prouve le résultat suivant. Etat donné une forme holomorphe cuspidale f et un point quelquonque du plan complexe, il existe une forme holomorphe cuspidale g tell...

Special Values of Abelian L - Functions at S = 0

In [12], Stark formulated his far-reaching refined conjecture on the first derivative of abelian (imprimitive) L–functions of order of vanishing r = 1 at s = 0. In [10], Rubin extended Stark’s refined conjecture to describe the r-th derivative of abelian (imprimitive) L-functions of order of vanishing r at s = 0, for arbitrary values r. However, in both Stark’s and Rubin’s setups, the order of ...

Random Matrix Theory and L - Functions at s = 1 / 2

Recent results of Katz and Sarnak [8,9] suggest that the low-lying zeros of families of L-functions display the statistics of the eigenvalues of one of the compact groups of matrices U(N), O(N) or USp(2N). We here explore the link between the value distributions of the L-functions within these families at the central point s = 1/2 and those of the characteristic polynomials Z(U, θ) of matrices ...

Nonvanishing of quadratic Dirichlet L - functions at s

The Generalized Riemann Hypothesis (GRH) states that all non-trivial zeros of Dirichlet L-functions lie on the line Re(s) = 12 . Further, it is believed that there are no Q-linear relations among the non-negative ordinates of these zeros. In particular, it is expected that L( 1 2 , χ) 6= 0 for all primitive characters χ, but this remains still unproved. This appears to have been first conjectur...