calculus and analysis - 3N-dimensional integral

I'm trying to compute a multidimensional integral with a variable number of dimensions.

The integral is as follows:

$\int d^{3N}\!p~e^{-\frac{\beta}{2m}\vec p^2}.$

I have tried this

Integrate[e^(-a*{p1,p2,p3}^2),{{p1,p2,p3}^N,-Infinity,Infinity}]

but it's not working at all. I guess I have to determine the vector before, or?

More generally, to what extent can Mathematica compute such integrals for arbitrary $N$ and arbitrary integrands in place of $e^{-\frac{\beta}{2m}\vec p^2}$ ?

Comments

functions - Get leading series expansion term?

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