  We study the Epstein zeta function $E_n(L,s)$ for $s>\frac{n}{2}$ and
determine for fixed $c>\frac{1}{2}$ the value distribution and moments of
$E_n(\cdot,cn)$ (suitably normalized) as $n\to\infty$. We further discuss the
random function $c\mapsto E_n(\cdot,cn)$ for $c\in[A,B]$ with $\frac{1}{2}<A<B$
and determine its limit distribution as $n\to\infty$.
