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 ♯ ♭ .
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