Value of polygamma at 1/2

Theorem

The following formula holds for $m=1,2,3,\ldots$: $$\psi^{(m)} \left( \dfrac{1}{2} \right) = (-1)^{m+1} m! \left( 2^{m+1}-1 \right) \zeta(m+1),$$ where $\psi^{(m)}$ denotes the polygamma, $m!$ denotes the factorial, and $\zeta(m+1)$ denotes the Riemann zeta.

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous): 6.4.4