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 Rs>σT(θ):=1/2+(logT)−θ and T<Ims<2T, to provide its asymptotic formula for fixed 0<θ<1 conditionally. Moreover, it is unconditional if the class number of Q is 2 or 3 and 0<θ<1/13.