  We investigate the zeros of Epstein zeta functions associated with a positive
definite quadratic form with rational coefficients in the vertical strip $
\sigma_1 < \Re s < \sigma_2 $, where $ 1/2 < \sigma_1 < \sigma_2 < 1 $. When
the class number of the quadratic form is bigger than 1, Voronin gives a lower
bound and Lee gives an asymptotic formula for the number of zeros. In this
paper, we improve their results by providing a new upper bound for the error
term.
