If you know about the Riemann hypothesis then you probably are aware of the critical strip. In short, I'm wondering how to "reverse-construct" a function based on the generalization of the critical strip that respects some additional symmetry requirements and that also satisfies its own sort of Riemann hypothesis.
We begin by generalizing the region from the real open unit interval $X=(0,1)$ to the region $X^2=(0,1)^2$ and a priori assuming that the infnite cuboid region for which $X^2$ is a slice, is an analytically continued region of some function we need to find. Insert a vertical imaginary line passing through the point $p \in X^2$ where $p=(1/2,1/2)$ and let this be the critical line for which the non-trivial zeros of our function should lie on.
The function we would like to construct should respect a functional equation that respects four symmetries (based on the symmetries of the square which has four lines of symmetry) as opposed to the case with the functional equation for the Riemann zeta function which satisfies the symmetry $\Psi(s)=\Psi(1-s)$ in the critical strip.
How do you construct such a function and does one already exist in the literature? Is it impossible for such a function to exist?
