. Let E be Galois extension of Q of finite degree and let π and π ′ be two irreducible automorphic unitary cuspidal representations of GL m ( E A ) and GL m ′ ( E A ), respectively. We prove an asymptotic formula for computation of coefficients γ π,π ′ ( k ) in the Laurent (Taylor) series expansion around s = 1 of the logarithmic derivative of the Rankin-Selberg L − function L ( s,π × e π ′ ) under assumption that at least one of representations π , π ′ is self-contragredient and show that coefficients γ π,π ′ ( k ) are related to weighted Selberg orthogonality. We also replace the assumption that at least one of representations π and π ′ is self-contragredient by a weaker one.

: We investigate relations between zero-free regions of certain L -functions and the asymptotic behavior of corresponding generalized Li coefficients. Precisely, we prove that violation of the (cid:28)= 2-generalized Riemann hypothesis implies oscillations of corresponding (cid:28) -Li coefficients with exponentially growing amplitudes. Results are obtained for class S ♯♭ ( (cid:27) 0 ; (cid:27) 1 ) that contains the Selberg class, the class of all automorphic L -functions, the Rankin{Selberg L functions, and products of suitable shifts of the mentioned functions.

The Li coefficients $\unicode[STIX]{x1D706}_{F}(n)$ of a zeta or $L$ -function $F$ provide an equivalent criterion for the (generalized) Riemann hypothesis. In this paper we define these coefficients, and their generalizations, the $\unicode[STIX]{x1D70F}$ -Li coefficients, for a subclass of the extended Selberg class which is known to contain functions violating the Riemann hypothesis such as the Davenport–Heilbronn zeta function. The behavior of the $\unicode[STIX]{x1D70F}$ -Li coefficients varies depending on whether the function in question has any zeros in the half-plane $\text{Re}(z)>\unicode[STIX]{x1D70F}/2.$ We investigate analytically and numerically the behavior of these coefficients for such functions in both the $n$ and $\unicode[STIX]{x1D70F}$ aspects.

We prove that there exists an entire complex function of order one and finite exponential type that interpolates the Li coefficients λF(n) attached to a function F in the class that contains both the Selberg class of functions and (unconditionally) the class of all automorphic L-functions attached to irreducible, cuspidal, unitary representations of GLn(ℚ). We also prove that the interpolation function is (essentially) unique, under generalized Riemann hypothesis. Furthermore, we obtain entire functions of order one and finite exponential type that interpolate both archimedean and non-archimedean contribution to λF(n) and show that those functions can be interpreted as zeta functions built, respectively, over trivial zeros and all zeros of a function .

We prove that H-invariants and the conductor of the function F belonging to the Selberg class can be represented as special values of (a meromorphic continuation of) a ”superzeta” function arising from non-trivial zeros of F .