The Epstein zeta-function $$\zeta (s; Q)$$ , $$s=\sigma +it$$ , is defined, for $$\sigma >\frac{n}{2}$$ , by the series $$\begin{aligned} \zeta (s; Q)=\sum _{{\underline{x}}\in {\mathbb {Z}}^n\setminus \{{\underline{0}}\}} (Q[{\underline{x}}])^{-s}, \end{aligned}$$ where Q is a positive definite quadratic $$n\times n$$ matrix and $$Q[{\underline{x}}]= {\underline{x}}^{\mathrm {T}} Q {\underline{x}}$$ , and is meromorphically continued to the whole complex plane. In the paper, we prove, for $$\sigma >\frac{n-1}{2}$$ , a limit theorem on the weak convergence of $$\begin{aligned} \frac{1}{T} \mathrm {meas} \left\{ t\in [0,T]: \zeta (\sigma +it; Q)\in A\right\} , \quad A\in {\mathcal {B}}({\mathbb {C}}), \end{aligned}$$ as $$T\rightarrow \infty $$ . The limit measure is given explicitly.