## More symmetry and Another Product

We’ve seen that $$\zeta(s)$$ satisfies the functional equation $\zeta(1-s)=2^{1-s}\pi^{-s}\cos(\pi s/2)\Pi(s-1)\zeta(s).$ (Well, it still needs to be proved, but let’s just assume it’s correct for now.) The goal of this post is an even more symmetrical form that will yield the function $$\xi(s)$$ which we can develop into an incredibly useful product expression. On our wish list for $$\xi(s)$$ we find three items: It’s an entire function, i.e., a function that’s holomorphic everywhere in $$\mathbb{C}$$ without any poles. [Read More]