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We have

$$\begin{align} &\Gamma\Big(1~+~x\Big)~\cdot~\Gamma\Big(1-x\Big)~=~\frac{\pi x}{\sin\pi x} \\\\ &\Gamma\Big(\tfrac12+x\Big)~\cdot~\Gamma\Big(\tfrac12-x\Big)~=~\frac\pi{\cos\pi x} \\\\ &\Gamma\Big(\tfrac14+x\Big){\bigg/}\Gamma\Big(\tfrac14-x\Big)~=~\pi^{2x}\cdot\frac{\zeta\Big(\tfrac12-2x\Big)}{\zeta\Big(\tfrac12+2x\Big)} \end{align}$$

where the latter is nothing else than the functional equation of the Riemann $\zeta$ function. I was

wondering whether yet another similar relation might also exist, involving either $\Gamma\Big(\tfrac18\pm x\Big)$

or $\Gamma\Big(\tfrac13\pm x\Big).$

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