Formula:8390

Digital Library of Mathematical Functions ID 25.11.E23

${\displaystyle{\displaystyle\zeta'\left(1-2n,\tfrac{1}{3}\right)=-\frac{\pi(9^{n}-1)B_{2n}}{8n\sqrt{3}(3^{2n-1}-1)}-\frac{B_{2n}\ln 3}{4n\cdot 3^{2n-1}}-\frac{(-1)^{n}{\psi^{(2n-1)}}\left(\frac{1}{3}\right)}{2\sqrt{3}(6\pi)^{2n-1}}-\frac{\left(3^{2n-1}-1\right)\zeta'\left(1-2n\right)}{2\cdot 3^{2n-1}},}}$

Constraint(s)

${\displaystyle{\displaystyle n=1,2,3,\dots}}$

Symbols List

• ${\displaystyle{\displaystyle B_{\NVar{n}}}}$:
• ${\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}$: Hurwitz zeta function
• ${\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}$: Riemann zeta function
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}$: psi (or digamma) function
• ${\displaystyle{\displaystyle\ln\NVar{z}}}$: principal branch of logarithm function
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
 

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