$\DeclareMathOperator\SL{SL}$What is the expected value of the length of the shortest non-zero vector in a (unimodular) lattice? I.e., let $G=\SL_n(\mathbb{R})$ with Haar measure $\mu$, $\Gamma=\SL_n(\mathbb{Z})$, $\lambda_1(\Lambda)$ the length of shortest non-zero vector in the lattice $\Lambda$, and consider the average $$ \frac{1}{\mu(G/\Gamma)}\int_{G/\Gamma}\lambda_1(\Lambda) \ d\mu(\Lambda). $$ For example in dimension two we get something like $$ \frac{3}{\pi}\int_{x=-1/2}^{x=1/2}\int_{y=\sqrt{1-x^2}}^{y=\infty}\frac{1}{\sqrt{y}}\frac{dydx}{y^2}\approx 0.682679. $$ I assume this is hard to calculate or approximate for general $n$ (fundamental domain, meaningful expressions for $\lambda_1$ given some representative basis for $\Lambda$, etc.).

I see things like the "Gaussian heuristic" in lattice crypto papers, but are there any reasonable results on such lattice statistics? Even numerical results (e.g. sample the space, approximate $\lambda_1$ on the samples, average) would be interesting.