A condition for the Riemann Zeta-function by modification of its functional equation 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: This Mathematica plot indicates $s=1/2$ is the only solution for $|{\rm Im}|\,s<2$.

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 ${\rm Re}\,s=1/2$:

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 ${\rm Re}\,s\neq 1/2$.

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
$y_0=0$, $y_1=15.1407$, $y_2=21.7412$, $y_3=25.9546$, $y_4=30.8933$, $y_5=33.931$, $y_6=38.1796$, $y_7=41.448$, $y_8=44.3018$, $y_9=48.2703$, $y_{10}=50.6212$.

