On asymptotic behavior of generalized Li coefficients in the Selberg class

In this paper we obtain a full asymptotic expansion of the archimedean contribution to the Li coefficients λF(−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 GLN(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 λF(−n), as n→+∞ equivalent to Generalized Riemann Hypothesis (GRH) true and GRH false for F∈S♯♭.