Explicit zero-free regions and a τ-Li-type criterion

 τ-Li coefficients describe if a function satisfies the GeneralizedRiemann Hypothesis or not. In this paper we prove that certain values of theτ-Li coefficients lead to existence or non-existence of certain zeros. The firstmain result gives explicit numbers N₁ and N₂ such that if all real parts ofthe τ-Li coefficients are non-negative for all indices between N₁ and N₂, thenthe function has non zeros outside a certain region. According to the secondresult, if some of the real parts of the τ-Li coefficients are negative for someindex n between numbers n₁ and n₂, then there is at least one zero outside acertain region.