Motivated by the analogy between spectral moments of random matrices and associated zeta functions, we study inverse power trace moments of the Laguerre ensemble of dimension $N$ and inverse temperature parameter $β>0$. We consider a large $N$ regime determined by the low-lying eigenvalues of the ensemble known as the hard edge. In the classical cases $β\in \{1,2,4\}$, we obtain explicit results for the inverse moments and extend these to formulae for the corresponding Mellin transforms. In the case of general $β>0$, by a result of Fyodorov and Le Doussal, we obtain a different formula for the moments given as a sum over partitions. We use this to consider a low temperature limit where $β\to \infty$ as $N \to \infty$. In this limit, we show that the moments are given in terms of the Bessel zeta function.