  For a fixed integer $l\geq 1$, let $R(t)$ denote the error term in the Weyl's
law of a $(2l+1)$-dimensional Heisenberg manifold with the metric $g_l.$ In
this paper we shall prove the asymptotic formula of the $k$-th power moment for
any integers $3\leq k\leq 9.$ We shall also prove that the function
$t^{-(l-1/4)}R(t)$ has a distribution function.
