On generalized Li criterion for a certain class of $L-$functions

We define generalized Li coefficients, called $τ-$Li coefficients for a very broad class $\mathcal{S}^{\sharp \flat }(σ_0, σ_1)$ of $L-$functions that contains the Selberg class, the class of all automorphic $L-$functions and the Rankin-Selberg $L-$functions, as well as products of suitable shifts of those functions. We prove the generalized Li criterion for zero-free regions of functions belonging to the class $\mathcal{S}^{\sharp \flat }(σ_0, σ_1)$, derive an arithmetic formula for the computation of $τ-$Li coefficients and conduct numerical investigation of $τ-$Li coefficients for a certain product of shifts of the Riemann zeta function.

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