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

This paper studies deformations of the well-known $\mathrm{\Xi }(s)=\mathrm{\Xi }(1-s)$ equation for the Riemann $\mathrm{\Xi }$ function satisfied by ${\mathrm{\Xi }}_{\rho }(s):={\int }_0^{\mathrm{\infty }}\frac{\text{d}t}{t}{t}^{\frac{s}{2}}\mathrm{\Psi }(t)e^{-\rho {\ln }^2(t)}$ where $\mathrm{\Psi }(t)={\sum }_{n\in {\mathbb{N}}^{+}}{e}^{-\pi n^{2}t}$.