On τ-Li-type criterion and explicit zero-free regions in the critical strip

τ-Li coefficients describe if a function satisfies the Generalized Riemann Hypothesis or not. In this paper we prove explicit conditions for the τ-Li coefficients to hold if a function has zeros in certain regions or if all zeros lie in certain regions of the critical strip. The first main result gives an explicit number N such that for a non-negative integer M some of the real parts of the τ-Li coefficients between indices N and 5NM is negative if the function has a zero outside a certain region. According to the second result if all zeros of the function lie in certain region then real parts of the τ-Li coefficients are non-negative for indices n∈[n1,n2].