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How to prove that

$$\zeta '(-1, 1/3)- \zeta '(-1, 2/3) = \frac{\psi ' (1/3)}{6 \sqrt{3} \pi}-\frac{\pi}{9 \sqrt{3}}$$

Does there exist a general formula for

$$\zeta '(-1, x)- \zeta '(-1, 1-x) $$

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