Riemann Xi function is defined as $$\xi(s)=\frac{s(s-1)}{2}\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$$ Question What is the derivative of $\xi(s)$ at $\frac{1}{2}$?
My try Logarithmic differentiation gives $$\frac{\xi'(s)}{\xi(s)}=\frac{1}{s}+\frac{1}{s-1}-\frac{log(\pi)}{2}+\frac{\Gamma'(\frac{s}{2})}{\Gamma(\frac{s}{2})}+\frac{\zeta'(s)}{\zeta(s)}$$ At, $s=\frac{1}{2}$ $$\frac{\xi'(\frac{1}{2})}{\xi(\frac{1}{2})}=-\frac{log(\pi)}{2}+\frac{\Gamma'(\frac{1}{4})}{\Gamma(\frac{1}{4})}+\frac{\zeta'(\frac{1}{2})}{\zeta(\frac{1}{2})}$$
$\xi(1/2+s)$ is even, $\xi'(1/2)=0$.