On A Li-Type Criterion For Zero-Free Regions Of Certain Dirichlet Series With Real Coefficients

The Li coefficients lambda(F)(n) of a zeta or L-function F provide an equivalent criterion for the (generalized) Riemann hypothesis. In this paper we define these coefficients, and their generalizations, the tau-Li coefficients, for a subclass of the extended Selberg class which is known to contain functions violating the Riemann hypothesis such as the Davenport-Heilbronn zeta function. The behavior of the tau-Li coefficients varies depending on whether the function in question has any zeros in the half-plane Re(z) > tau/2. We investigate analytically and numerically the behavior of these coefficients for such functions in both the n and tau aspects.