Problem statement: While I was writing another question and explaining, what I have already tried so far, I came across the following identity: $$\frac{\Gamma\left(\frac{1-2\,s}{2}\right)\,\zeta\left(-2\,s\right)}{\sqrt{\pi}\,s}=-\frac{\pi^{-2s}\,\sec\left(\pi\,s\right)\,\zeta\left(1+2\,s\right)}{\Gamma\left(1-s\right)}$$ $\zeta\left(s\right)$ is the Riemann zeta-function and $\Gamma\left(s\right)$ the gamma function. I assume, that this identity can be proven by the Zeta functional equation, but I haven't figured out how to show that.
EDIT 29.05.21
Inspired by a recent comment of @StevenClark and @metamorphy I finally managed to derivate the reflection functional equation of the Riemann zeta-function starting from the identity above. For the proof only the substituion $s\to -\frac{s}{2}$ is needed.
Attempt at an alternative proof: I found the following alternative proof, which is very interesting. We rewrite the identity in the form: $$-2\frac{\Gamma\left(\frac{1-2\,s}{2}\right)}{\sqrt{\pi}\,s}\sum_{k=1}^{\infty}\left(\frac{1}{k}\right)^{-2s}=\frac{4}{\pi^{\frac{3}{2}}}\frac{\pi^{\frac{3}{2}-2s}\sec\left(\pi\,s\right)}{2\,\Gamma\left(1-s\right)}\sum_{k=1}^{\infty}\left(\frac{1}{k}\right)^{1+2s}$$ Using the Mellin inverse transform on both sides leads to the interesting identity: $$1+2\sum_{k=1}^{\infty}\operatorname{erfc}\left(\frac{k}{\sqrt{z}}\right)=\frac{2\sqrt{z}}{\sqrt{\pi}}+\frac{4}{\pi^{\frac{3}{2}}}\sum_{k=1}^{\infty}\frac{\text{DawsonF}[k\,\pi\,\sqrt{z}]}{k}$$ $\text{DawsonF}\left(z\right)$ is Dawson's integral and $\text{erfc}\left(z\right)$ is the complementary error function. The left- and right-hand side may be written by its integral representation: $$\mathcal{G}\left(z\right)=\frac{1}{\pi}\int_{0}^{1}\frac{\vartheta_{3}\left(0,\exp\left(-\frac{1}{z~\tau}\right)\right)}{\sqrt{\tau\,\left(1-\tau\right)}}=\sqrt{\frac{z}{\pi}}\int_{0}^{1}\frac{\vartheta_{3}\left(0,\exp\left(-\pi^{2}\,z~\tau\right)\right)}{\sqrt{\left(1-\tau\right)}}$$ The equality may be proved by the Jacobi theta functional equation Peter Woit,p.4. The new function $\mathcal{G}\left(z\right)$ , will be subject of a future post of us. It has the known property of the theta function $\vartheta_{3}$, too: the functional equation. The proof of this transformation formula is then based on a result from theory of Fourier series called Poisson Summation Formula. For this the Fourier transform has to be applied, resulting in the last identity.