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:
