Gaussian Kernel Expectation

Question

Let $\kappa(x, x')=\exp(-\frac{1}{2\sigma^2}||x-x'||)$ be the Gaussian kernel with the multivariate random variable $x, x' \in \mathbb{R}^D$ and scalar parameter $\sigma \in \mathbb{R}_+$. I am interested in the expectation $\mathbb{E}_{x,x'} [\kappa(x,x')]$ with $p(x)=p(x')=\mathcal{N}(\mu, \Sigma)$. Is there any closed form solution to it?