Integral involving $L_2$-norm Consider the following integral:
$$
\int_{[0,1]^4} e^{-\|x\|_2\cdot\sqrt\frac{2\pi n}{\lambda}}\,dx
$$
Is there a closed-from expression for this integral?

The integral comes from small $x$. The result is (here $d=4$):
$$
\frac{1}{2^d}\frac{2 \pi ^{d/2}}{ \Gamma \left(\frac{d}{2}\right)} \int_0^{\infty}\rho ^{d-1} \exp \left(-\rho  \sqrt{\frac{2 \pi  n}{\lambda }}\right)d\rho=\frac{3 \lambda ^2}{16 n^2}.
$$
