An Explicit Formula and Estimations for Hecke L-Functions: Applying the Li Criterion

We advance the use of a criterion for the validity of the general-ized Riemann hypothesis. We rewrite and then estimate various terms of an explicit formula for Hecke L-functions very recently obtained by Li. We demonstrate how additional information on a coefficient func-tion appearing in the logarithmic derivative of the L function would lead directly to verification of a generalized Riemann hypothesis. More generally, we mention a direct link between the generalized Ramanujan conjectures and the validity of generalized Riemann hypotheses