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

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We define generalized Li coefficients, called τ-Li coefficients for a very broad class S (σ₀, σ₁) 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 S (σ₀, σ₁), 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.