Zeros of the Epstein zeta-function

I shall talk about the proof of the following result: the critical line contains a positive proportion of non-trivial zeros of the Epstein zeta-function, corresponding to a binary positive definite quadratic form. The proof of this result for the function which violates the Riemann hypothesis (it has many non-trivial zeros off the critical line) relies heavily on the methods developed by Atle Selberg. We shall discuss some results, which can be obtained in the theory of L-functions following Selberg's ideas.