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$\tau$-Li coefficients describe if a function satisfies the GeneralizedRiemann Hypothesis or not. In this paper we prove that certain values of the$\tau$-Li coefficients lead to existence or non-existence of certain zeros. Thefirst main result gives explicit numbers $N_1$ and $N_2$ such that if all realparts of the $\tau$-Li coefficients are non-negative for all indices between$N_1$ and $N_2$, then the function has non zeros outside a certain region.According to the second result, if some of the real parts of the $\tau$-Licoefficients are negative for some index $n$ between numbers $n_1$ and $n_2$,then there is at least one zero outside a certain region.

Explicit zero-free regions and a $\tau$-Li-type criterion