The paper deals with the functions \(f_{\mu, \lambda}, \lambda > 0,\) defined by \[ f_{\mu, \lambda}(z) =\int_{-\infty}^\infty \exp\left(-\frac{\lambda}{2}t^2+izt\right)d\mu(t), \] where \(\mu\) is a complex Borel measure, satisfying \( \mu (-E) =\overline{\mu(E)}\) for every Borel set \(E\subset \mathbb{R}.\) The authors investigate the asymptotic behavior of the zero distribution of \(f_{\mu, \lambda}\) for \(\lambda \to \infty.\) In particular they obtain the complete description of all complex Borel measures \(\mu,\) such that the functions \(f_{\mu, \lambda}\) have only real zeros. This result is the generalization of Newman's results. Theorems of this paper are applied to the Riemann \(\zeta\)-function.