Let $W$ be the normalized spherical Whittaker function attached to a spherical representation $\pi$ on $GL_n(k)$, where $k$ is a $p$-adic field and $n\ge 3$.

I'm faced with the slightly-odd integral $$\int_{k^\times} \left|W\left(\matrix{y&&&\cr &\ddots&&\cr &&y&\cr &&&1}\right)\right|^2 |y|^s\ dy$$ where ${\rm Re}(s)$ is sufficiently large to get convergence.

Does anyone have any ideas about calculating this?

Recall the Casselman-Shalika-Shintani formula, $$W(\varpi^J)=\delta_B^{1/2}(\varpi^J){\rm Tr}\left(\rho_J\left(A_\pi\right)\right)$$ where $J=(j_1,\ldots,j_n)\in\mathbb Z^n$ satisfies $j_1\ge j_2\ge\ldots\ge j_n$, $\varpi^J$ is the diagonal matrix with $j$-th entry $\varpi^{j_i}$, $\delta_B$ is the modular character of the Borel subgroup, $\rho_J$ is the representation of $GL_n(\mathbb C)$ with highest weight $(j_1,\ldots,j_n)$, and $A_\pi$ is the matrix of Satake parameters of $\pi$. Using this, the integral can be written as $$\sum_{i\ge 0} q^{-i(s+n-1)}{\rm Tr}\left(\rho_{(i,\ldots,i,0)}(A_\pi)\otimes\rho_{(i,\ldots,i,0)}(\bar A_\pi)\right)$$ where the shift by $n-1$ comes from $\delta_B(\varpi^{(i,\ldots,i,0)})=\prod_{j=0}^{n-2}|\varpi^i|^{n-1-2j}$.

This is basically a sub-sum of the formula used to get the Rankin-Selberg $L$-function for $\pi\otimes\tilde\pi$, so it should have a reasonable answer.