The analytic continuation conjecture for complete adjoint L-functions

Let $F$ be a global field, let $\pi$ be a unitary cuspidal representation of $\operatorname{GL}(n,\mathbb{A}_F)$, and let $\tilde{\pi}$ be its contragredient. Define the complete adjoint $L$-function by

$$\Lambda(s,\pi,\operatorname{Ad})=\frac{\Lambda(s,\pi\times\tilde{\pi})}{\Lambda_F(s)}.$$

Adjoint $L$-function analytic continuation conjecture. The complete adjoint $L$-function $\Lambda(s,\pi,\operatorname{Ad})$ admits an analytic continuation to the whole complex plane.

This is a basic consequence expected from the Langlands program, identifying the adjoint $L$-function with the Langlands $L$-function for the adjoint action on $\mathfrak{sl}(n,\mathbb{C})$. The paper proves entireness of complete adjoint $L$-functions for unitary cuspidal representations of $\operatorname{GL}(3)$ and $\operatorname{GL}(4)$, but the general statement remains open.