Let $Q$ be a positive definite quadratic form with integral coefficients and let $E(s,Q)$ be the Epstein zeta function associated with $Q$. Assume that the class number of $Q$ is bigger than 1. Then we estimate the number of zeros of $E(s,Q)$ in the region $mathfrak{R}s>sigma_{T}(theta):=1/2 +(log T)^0$ and $T