The equation, $s\in\mathbb{C}$ with $0<\Re(s)<1$: $$\frac{\zeta(2-s)}{\zeta(1+s)}=\frac{\Gamma(1+\frac{s}{2})}{\Gamma(1+\frac{1-s}{2})}\pi^{\frac{1}{2}-s}$$ A general question: For which values the equation holds ? (Trivial case: $s=\frac{1}{2}$.)
Because I have not the technical possibility for a numerical evaluation, I have to ask: If $s$ is e.g. the first nontrivial zero of $\zeta(s)$, we get what difference between the left and the right ?
A second question: When do we get $$|\frac{\zeta(2-s)}{\zeta(1+s)}|<|\frac{\Gamma(1+\frac{s}{2})}{\Gamma(1+\frac{1-s}{2})}\pi^{\frac{1}{2}-s}|$$ ?
E D I T :
$$g(s):=\frac{\zeta(2-s)}{\zeta(1+s)}-\frac{\Gamma(\frac{1+s}{2})}{\Gamma(\frac{2-s}{2})}\pi^{\frac{1}{2}-s}$$
Can someone create plots/graphs and a list of values for $g(s)$ like Carlo Beenakker has done very good for $f(s)$ ? (the definition of $f(s)$ is below)
I like to know, if $f(s)$ was something special or if we get a similar interesting behavior of $\Im(s)$ for $g(s)=0$. Thank you !