We define modified Li coefficients, called τ -Li coefficients for a very broad class S (σ 0 , σ 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 S (σ 0 , σ 1 ), derive an arithmetic formula for the computation of τ -Li coefficients and conduct numerical investigation of τ -Li coefficients for a certain product of shifts of the Riemann zeta function. .fi (A.-M. Ernvall-Hytönen), almasa@pmf.unsa.ba (A. Odžak), lejlas@pmf.unsa.ba (L. Smajlović), medina.susic@pmf.unsa.ba (M. Sušić).

We define a generalized Li coefficient for the L-functions attached to the Rankin-Selberg convolution of two cuspidal unitary automorphic representations π and π of GL m (A F ) and GL m (A F ). Using the explicit formula, we obtain an arithmetic representation of the nth Li coefficient λ π,π (n) attached to L(s, π f × π f ). Then, we deduce a full asymptotic expansion of the archimedean contribution to λ π,π (n) and investigate the contribution of the finite (non-archimedean) term. Under the generalized Riemann hypothesis (GRH) on non-trivial zeros of L(s, π f × π f ), the nth Li coefficient λ π,π (n) is evaluated in a different way and it is shown that GRH implies the bound towards a generalized Ramanujan conjecture for the archimedean Langlands parameters μ π (v, j ) of π. Namely, we prove that under GRH for L(s, π f × π f ) one has | Re μ π (v, j )| ≤ 1 4 for all archimedean places v at which π is unramified and all j = 1, . . . , m.

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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 .

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