# Evaluation of Tornheim's type of double series

## TL;DR

This paper investigates specific Tornheim's double series with odd weight, confirming a conjecture related to the parity theorem for the zeta function of the exceptional Lie algebra G_2, advancing understanding in this mathematical area.

## Contribution

It provides an affirmative proof of a conjecture about the parity theorem for the G_2 zeta function using Tornheim's series analysis.

## Key findings

- Confirmed the parity theorem conjecture for G_2 zeta function
- Derived explicit values for Tornheim's series with odd weight
- Enhanced understanding of zeta functions related to Lie algebras

## Abstract

We examine values of certain Tornheim's type of double series with odd weight. As a result, an affirmative answer to a conjecture about the parity theorem for the zeta function of the root system of the exceptional Lie algebra $G_2$, proposed by Komori, Matsumoto and Tsumura, is given.