A p-adic integral

Question

Let $(K,||)$ be a finite extension of $\mathbb{Q}_p$ of degree $d$ such that the restriction of $||$ to $\mathbb{Q}_p$ is the usual p-adic absolute value. Endow $GL_n(K)$ with the unique Haar measure $d^{\times} \mu$ which gives mass $1$ to the compact subgroup $K = GL_n(\mathcal{O}_K) = SL_n(K) \cap M_n( \mathcal{O}_K )$. My question is the following : does the integral $$ I(s) = \int_{GL_n(K)} \mathrm{1}_{\psi \in M_n( \mathcal{O}_K )} |\det \psi|^{ds} d^{\times} \mu(\psi) $$ have an "explicit" expression ? The case $n=1$ is easy (if $q$ is the cardinality of the residue field of $K$, then $I = (1-q^{-s})^{-1}$) and I didn't succeeded in calculating $I(s)$ for general $n$.