Archimedean zeta integrals on $G{L}_n\times G{L}_m$ and $S{O}_{2n+1}\times G{L}_m$

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