Abstract. In this paper, we consider the asymptotic behaviour of τ -Li coefficients for the wide class of L-functions that contains the Selberg class, the class of all automorphic L-functions, the Rankin-Selberg L-functions, as well as products of suitable shifts of the mentioned functions. We consider both archimedean and non-archimedean contribution to the τ -Li coefficients, both separately, and their joint contribution to the coefficients. We also derive the behavior of the coefficients in the case the τ/2-Riemann hypothesis holds, which is the generalization of the Riemann hypothesis for the class under consideration. Finally, we conclude with some examples and numerics.

. Let E be Galois extension of Q of finite degree and let π and π ′ be two irreducible automorphic unitary cuspidal representations of GL m ( E A ) and GL m ′ ( E A ), respectively. We prove an asymptotic formula for computation of coefficients γ π,π ′ ( k ) in the Laurent (Taylor) series expansion around s = 1 of the logarithmic derivative of the Rankin-Selberg L − function L ( s,π × e π ′ ) under assumption that at least one of representations π , π ′ is self-contragredient and show that coefficients γ π,π ′ ( k ) are related to weighted Selberg orthogonality. We also replace the assumption that at least one of representations π and π ′ is self-contragredient by a weaker one.

We define generalized Li coefficients, called $\tau-$Li coefficients for a very broad class $\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)$ of $L-$functions that contains the Selberg class, the class of all automorphic $L-$functions and the Rankin-Selberg $L-$functions, as well as products of suitable shifts of those functions. We prove the generalized Li criterion for zero-free regions of functions belonging to the class $\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)$, derive an arithmetic formula for the computation of $\tau-$Li coefficients and conduct numerical investigation of $\tau-$Li coefficients for a certain product of shifts of the Riemann zeta function.

Abstract In this paper we obtain a full asymptotic expansion of the archimedean contribution to the Li coefficients λ F ( − n ) (n is a positive integer) attached to a function F in the certain class S ♯ ♭ of functions containing the Selberg class S and (unconditionally) the class of all automorphic L-functions attached to irreducible, unitary cuspidal representations of GL N ( Q ) . Applying the obtained results to automorphic L-functions, we improve the result of J.C. Lagarias concerning the asymptotic behavior of archimedean contribution to the nth Li coefficient attached to the automorphic L-function. We also deduce asymptotic behaviors of λ F ( − n ) , as n → + ∞ equivalent to Generalized Riemann Hypothesis (GRH) true and GRH false for F ∈ S ♯ ♭ .