  Let $\Pi$ be a cuspidal automorphic representation of
$\mathrm{GL}_5(\mathbb{A}_\mathbb{Q})$, and let $\Sigma = \mathrm{Ind}(\pi,
\chi_2|\cdot|^{1/2}, \chi_3|\cdot|^{-1/2})$ be an automorphic representation of
$\mathrm{GL}_4(\mathbb{A}_\mathbb{Q})$ induced from the standard parabolic
subgroup of the form $(2, 1, 1)$ where $\pi$ is a cuspidal automorphic
representation of $\mathrm{GL}_2(\mathbb{A}_\mathbb{Q})$. Assume that
$\Pi_{\infty}$ and $\Sigma_{\infty}$ are cohomological with respect to the
trivial representation in which case $s=1/2$ is a critical point for the
Rankin-Selberg $L$-function $L(s, \Pi \times \Sigma)$. Following Mahnkopf, we
prove a result about $L(\tfrac{1}{2}, \Pi \times \Sigma)$, and as a corollary,
obtain an algebraicity result for the ratio $L(1, \Pi \times \chi)/L(1, \Pi
\times \chi')$, where $\chi,\chi'$ are finite order Hecke characters such that
$\chi_{\infty} = \chi'_{\infty} = \mathrm{sgn}$.
