On the modified Li criterion for a certain class of L-functions

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