On the computation of local adjoint $L$-function of unramified representation

Question

Let $F$ be a p-adic local field and $W$ be a 2n-dimensional symplectic space over $F$. Let $G_n$ be the isometry group of $W$ and $B_n$ be the Borel subgroup of $G_n$. Then the maximal torus $T_n$ of $B_n$ is isomorphic to $(GL_1)^n$. Let $\pi$ be $Ind_{B_n(F)}^{G_n(F)} (\chi_1 \boxtimes \cdots \boxtimes \chi_n)$, the normalized induction of the character $\chi_1 \boxtimes \cdots \boxtimes \chi_n$ of $T_n$ to $G_n$. Can we express the adjoint $L$-function $L(s,\pi,Ad)$ in terms of $\chi_1,\cdots,\chi_n$?