Meromorphic structure of the Mellin family L_k and its parameter derivative at k=1 / Estrutura meromorfa da família de Mellin L_k e sua derivada paramétrica em k=1

This revised preprint studies L_k(s)=integral_0^infinity t^(s-1)/(exp(t+t^k)-1) dt for k>0. It proves a meromorphic continuation through generalized-power expansions and gives a finite formula for candidate residues, including a universal simple pole at s=0 for k not equal to 1. It provides exact residues for k=2, covers 0<k<1, and proves that the parameter derivative at k=1, initially for Re(s)>1, equals -(1/2)H'(s), where H(s)=2^(-s)Gamma(s+1)zeta(s). The paper distinguishes the nontrivial zeros of the Riemann zeta function from critical points of H and provides reproducible numerical examples. No proof of the Riemann hypothesis or claim of bibliographically certified novelty is made.