On the number of local minima of integer lattices

Let $E_s(N)$ be the average number of local minima of $s$-dimensional integer lattices with determinant equals $N$. We prove the following estimates
$$ \frac{2^{-1}}{(s-1)!}+O_s\left(\frac{1}{\ln N}\right)\le\frac{E_s(N)}{\ln^{s-1}N}\le\frac{2^s}{(s-1)!}+O_s\left(\frac{1}{\ln N}\right) $$
for any prime $N$. Using this result we have a new lower bound for maximum number of local minima of integer lattices.