On the speiser equivalent for the riemann hypothesis

A. Speiser showed that the Riemann hypothesis is equivalent to the absence of non-trivial zeros of the derivative of the Riemann zeta-function left of the critical line. The quantitative version of this result was obtained by N. Levinson and H. Montgomery. This result (or the quantitative version of this result proved by N. Levinson and H. Montgomery) were generalized for many zeta-functions for which the Riemann hypothesis is expected. Here we generalize the Speiser equivalent for zeta-functions. We also investigate the relationship between the on-trivial zeros of the extended Selberg class functions and of their derivatives in this region. This class contains zeta functions for which Riemann hypothesis is not true. As an example, we study the relationship between the trajectories of zeros of linear combinations of Dirichlet $L$-functions and of their derivatives computationally