On the boundedness of Euler-Stieltjes constants for the Rankin-Selberg \(L-\)function

Let \(E\) be a Galois extension of \(Q\) of finite degree and let \(π \) and \(π'\) be two irreducible automorphic unitary cuspidal representations of \(GL_m(A_E)\) and \(GLm'(A_E)\), respectively. Let \(Λ(s,π×{π}')\) be a Rankin-Selberg \(L-\)function attached to the product \(π×{π}'\), where \({π}'\) denotes the contragredient representation of \(π'\), and let its finite part (excluding Archimedean factors) be \(L(s,π×{π}')\). The Euler-Stieltjes constants of the Rankin-Selberg \(L-\)function are the coefficients in the Laurent (Taylor) series expansion around \(s=1+it_0\) of the function \(L(s, π × {π}')\). In this paper, we derive an upper bound for these constants.