Given an integer $d$, let $\alpha_d(N)$ denote the number of symmetric integral positive definite matrices of size $d\times d$ with coefficients in $\lbrace -N,-N+1,\dots,N-1,N\rbrace$.

Asymptotically, the number $\alpha_d(N)$ is given by $\alpha_d(N)\sim c_dN^{d+1\choose 2}$ for some strictly positive real number $c_d$ (with $c_d\leq 2^{d+1\choose 2}$).

What is known on the constants $c_2,c_3,\dots$?