Laplace transform of the distance between 2 random points in a unit square

Question

Suppose that $X$ and $Y$ are independent uniformly distributed random vectors in $[0,1]^2$. Is it possible to derive a closed form expression of the Laplace transform of $\|X-Y\|_2$: $$ \mathsf{E}e^{-s\|X-Y\|_2}=? $$ When $s=\sqrt{2\pi b}$, Monte Carlo simulations suggest that $$ \mathsf{E}e^{-\|X-Y\|_2\sqrt{2\pi b}}\approx b^{-1} $$ for large $b$. Is it possible to show this "asymptotic" result at least?