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In this paper, we evaluate archimedean zeta integrals for automorphic$L$-functions on $GL_n \times GL_{n-1+\ell}$ and on $ SO_{2n+1} \timesGL_{n+\ell}$, for $\ell=-1$, $0$, and $1$. In each of these cases, the zetaintegrals in question may be expressed as Mellin transforms of products ofclass one Whittaker functions. Here, we obtain explicit expressions for theseMellin transforms in terms of Gamma functions and Barnes integrals.

Archimedean zeta integrals on $GL_n \times GL_m$ and $SO_{2n+1} \times GL_m$