Deformations of $\Xi (s)=\Xi (1-s)$ and the heat equation

Abstract

This paper studies deformations of the well-known $\Xi (s)=\Xi (1-s)$ equation for the Riemann $\Xi$ function satisfied by $Ξ_ρ(s):=\int_{0}^\infty\frac{{\d}t}{t}{t^{\frac{s}{2}}}Ψ(t)e^{-ρ\ln^2(t)}$ where $\Psi (t)={\sum }_{n\in {\mathbb{N}}^{+}}{e}^{-\pi n^{2}t}$.