  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} \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.
