Let $\zeta$ be the Riemann zeta function and $\beta$ the Dirichlet beta function. We know that $\zeta (0)=-1/2$ and $\beta (0)=1/2$ are algebraic numbers over $\mathbb{Q}$. This led me to the following question:
Are $\zeta'(0)$ and $\beta'(0)$ algebraic numbers?
It is known that $$\zeta'(0)=\log\frac{1}{\sqrt{2\pi}},$$ $$\beta'(0)=\log\frac{\Gamma (1/4)^2}{2\sqrt{2}\pi}$$ and that $\pi$ and $\Gamma (1/4)$ are algebraically independent over $\mathbb{Q}$ where $\Gamma$ is the gamma function.
Searching the decimal expansions of these constants on OEIS didn't yield anything, so I'm writing this question.
