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Is this equation for the Riemann zeta function discovered or re-discovered?

$$\left(\frac{\zeta(s)}{\zeta(1-s)}\right)^2 = (2 \cdot \pi)^{2s-1} \cdot \tan\left(\frac{\pi s}{2}\right) \cdot \frac{\Gamma(1-s)}{\Gamma(s)}$$

Where I derived it step by step (here, pages 21-22) and double-checked in Wolfram Alpha.

The principle root and second root after rearranging are incorrect:

$$\zeta(s) \ne \pm \zeta(1-s) \cdot (2 \cdot \pi)^{s-\frac{1}{2}} \cdot \sqrt{\tan\left(\frac{\pi s}{2}\right) \cdot \frac{\Gamma(1-s)}{\Gamma(s)}}$$

But the relative error between the left and right sides on the real axis is:

principle root second root

mattrdowney
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    As you've noted yourself, this equation is straightforward to derive from Riemann's functional equation and the gamma reflection formula. Does your equation have some particular significance or usefulness? By the way, what do you mean by "the relative error of the principal root and second root"? – Tanner Swett Sep 03 '23 at 00:56
  • I did not explain that well, I meant taking the square root of both sides of the equation and rearranging. I will edit my post. – mattrdowney Sep 03 '23 at 01:25
  • With usefulness, I have been trying to relate it to the product logarithm because the error at the critical strip [0,1] is half the length of the other error intervals, and the principle branch/sheet of the product logarithm seems to work similarly. – mattrdowney Sep 03 '23 at 01:57
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    If Wolfram Alpha verifies it as true, doesn't that imply that in a sense its already known? – Steven Clark Sep 03 '23 at 03:37

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