  Let $G={\rm GL}_{2n}$ over a totally real number field $F$ and $n\geq 2$. Let
$\Pi$ be a cuspidal automorphic representation of $G(\mathbb A)$, which is
cohomological and a functorial lift from SO$(2n+1)$. The latter condition can
be equivalently reformulated that the exterior square $L$-function of $\Pi$ has
a pole at $s=1$. In this paper, we prove a rationality result for the residue
of the exterior square $L$-function at $s=1$ and also for the holomorphic value
of the symmetric square $L$-function at $s=1$ attached to $\Pi$. On the way, we
also show a rationality result for the residue of the Rankin--Selberg
$L$-function at $s=1$, which is very much in the spirit of our recent joint
paper with Harris and Lapid, as well as of one of the main results in a recent
article of Balasubramanyam--Raghuram.
