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 ! 



A condition for the Riemann Zeta-function by modification of its functional equation
This Mathematica plot indicates $s=1/2$ is the only solution for $|{\rm Im}|\,s<2$.
![](https://ilorentz.org/beenakker/MO/contourplot.png)


Plotted versus ${\rm Re}\,s\in(0,1)$ and ${\rm Im}\,s\in(-2,2)$ is the absolute value $|f(s)|$ of the function
$$f(s)=\frac{\zeta(2-s)}{\zeta(1+s)}-\frac{\Gamma(1+\frac{s}{2})}{\Gamma(1+\frac{1-s}{2})}\pi^{\frac{1}{2}-s}$$


Further inspection shows oscillations of $|f(s)|$ for larger ${\rm Im}\,s$, which reach zero for ${\rm Re}\,s=1/2$:


![](https://ilorentz.org/beenakker/MO/zerosrealsonehalf.png)


The location of the zeroes is at $s\_n=1/2+iy\_n$ with


$y\_0=0$, $y\_1=14.2307$, $y\_2=20.9902$, $y\_3=25.1302$, $y\_4=30.2624$, $y\_5=33.1423$, $y\_6=37.5162$, $y\_7=40.8477$, $y\_8=43.5456$, $y\_9=47.7187$, $y\_{10}=49.975$.


I could not find any zeroes for ${\rm Re}\,s\neq 1/2$.




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ADDENDUM:
the plots for 
$$g(s)=\frac{\zeta(2-s)}{\zeta(1+s)}-\frac{\Gamma(\frac{1}{2}+\frac{s}{2})}{\Gamma(1-\frac{s}{2})}\pi^{\frac{1}{2}-s},$$
requested by the OP, are very similar to those of $f(s)$ (see below) with zeroes at
$y\_0=0$, $y\_1=15.1407$, $y\_2=21.7412$, $y\_3=25.9546$, $y\_4=30.8933$, $y\_5=33.931$, $y\_6=38.1796$, $y\_7=41.448$, $y\_8=44.3018$, $y\_9=48.2703$, $y\_{10}=50.6212$.


![](https://ilorentz.org/beenakker/MO/gsplots.png)


