  In this paper, we study the order of the pole of the triple tensor product
$L$-functions $L(s,\pi_1\times\pi_2\times\pi_3,\otimes^3)$ for cuspidal
automorphic representations $\pi_i$ of $\mathrm{GL}_{n_i}(\mathbb{A}_F)$ in the
setting where one of the $\pi_i$ is a monomial representation. In the view of
Brauer theory, this is a natural setting to consider. The results provided in
this paper give crucial examples that can be used as a point of reference for
Langlands' beyond endoscopy proposal.
