Published online by Cambridge University Press: 20 November 2018
Inspired by a construction by Bump, Friedberg, and Ginzburg of a two-variable integral representation on $\text{GS}{{\text{p}}_{4}}$ for the product of the standard and spin
$L$-functions, we give two similar multivariate integral representations. The first is a three-variable Rankin-Selberg integral for cusp forms on
$\text{PG}{{\text{L}}_{4}}$ representing the product of the
$L$-functions attached to the three fundamental representations of the Langlands
$L$-group
$\text{S}{{\text{L}}_{\text{4}}}\left( \text{C} \right)$. The second integral, which is closely related, is a two-variable Rankin-Selberg integral for cusp forms on
$\text{PGU}\left( 2,\,2 \right)$ representing the product of the degree
$8$ standard
$L$-function and the degree
$6$ exterior square
$L$-function.