TITLE: How are values of the Dirichlet Beta function derivative derived?
QUESTION [8 upvotes]: Wolfram Mathworld gives the following values for the beta function derivative. 
$$\beta'(-1) = \frac{2K}{\pi},\quad \beta'(0) = \ln \left[\frac{\Gamma^{2}(\frac{1}{4})}{2\pi\sqrt{2}} \right],\quad \beta'(1) =  \frac{\pi}{4}\left(\gamma +2\ln 2+3\ln\pi-4\ln \left(\Gamma \left(\frac{1}{4}\right) \right)\right)$$
I can see how $\beta'(1)$ could be derived from $\beta'(0)$ using the functional equation (or vice versa) but how was one of them found originally. The same for $\beta'(-1)$ and $\beta'(2)$. Is there any other known values of the derivative? Also can I have links to any papers on the subject, I cannot seem to find any?

REPLY [5 votes]: Another way to find $\beta'(1)$ is from Kummer's Fourier series for the logarithm of the gamma function:
$$ \log \Gamma(x) = \left(\tfrac{1}{2}-x \right)(\gamma+\log2)+(1-x)\log \pi - \frac{1}{2} \log \sin(\pi x) + \frac{1}{\pi} \sum_{n=1}^{\infty} \frac{\log n}{n} \sin(2\pi nx) $$
valid for $0<x<1 $. Setting $x=\frac{1}{4}$ gives
$$ \log \Gamma \left(\tfrac{1}{4} \right) = \frac{1}{4} (\gamma+\log 2) +\frac{3}{4}\log \pi -\frac{1}{2} \log \left(\tfrac{1}{\sqrt{2}} \right) +\frac{1}{\pi} \sum_{n=0}^{\infty} \frac{(-1)^n \log(2n+1)}{2n+1} $$
The sum appearing on the right is $-\beta'(1)$.