Let $$extract_itex$$Q$$/extract_itex$$ be a positive definite quadratic form with integral coefficients and let $$extract_itex$$E(s,Q)$$/extract_itex$$ be the Epstein zeta function associated with $$extract_itex$$Q$$/extract_itex$$. Assume that the class number of $$extract_itex$$Q$$/extract_itex$$ is bigger than 1. Then we estimate the number of zeros of $$extract_itex$$E(s,Q)$$/extract_itex$$ in the region $$extract_itex$$\mathfrak{R}s>\sigma_{T}(\theta):=1/2 +(\log T)^{-\theta}$$/extract_itex$$ and $$extract_itex$$T<Im s<2T$$/extract_itex$$, to provide its asymptotic formula for fixed $$extract_itex$$0<\theta<1$$/extract_itex$$ conditionally. Moreover, it is unconditional if the class number of $$extract_itex$$Q$$/extract_itex$$ is 2 or 3 and $$extract_itex$$0<\theta<1/13.$$/extract_itex$$