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:

  1. It's  an entire function, i.e., a function that's holomorphic everywhere in \mathbb{C} without any poles.
  2. It has zeros for all non-trivial zeros of the \zeta-function, and no others.
  3. It's perfectly symmetrical along the critical line, i.e., it satisfies \xi(1-s)=\xi(s).

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