Symmetry of zeros of Lerch zeta-function for equal parameters

Abstract

For most values of parameters $\lambda$ and $\alpha$, the zeros of the Lerch zeta-function $L(\lambda ,\alpha ,s)$ are distributed very chaotically. In this paper we consider the special case of equal parameters $L(\lambda ,\lambda ,s)$ and show by calculations that the nontrivial zeros either lie extremely close to the critical line $\sigma =1/2$ or are distributed almost symmetrically with respect to the critical line. We also investigate this phenomenon theoretically.

10 pages