# Symmetry of zeros of Lerch zeta-function for equal parameters

## TL;DR

This paper investigates the zeros of the Lerch zeta-function when parameters are equal, revealing near-symmetry and critical line proximity, supported by calculations and theoretical analysis.

## Contribution

It demonstrates the symmetry properties of zeros for the special case of equal parameters in the Lerch zeta-function, combining computational and theoretical approaches.

## Key findings

- Zeros are nearly symmetric with respect to the critical line.
- Nontrivial zeros lie very close to the critical line.
- Theoretical explanation supports computational observations.

## 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.