In this question Agno asked about the zeros of $\zeta\left(\frac{s}{a}\right) \pm \zeta\left(\frac{1-s}{a}\right)$.

I fixed $a=2$ and the minus sign and defined:

$$ f(s)=\Re \left( \zeta\left(\frac{s}{2}\right) - \zeta\left(\frac{1-s}{2}\right)\right) $$

The X-Ray of $f(s)$ looks like this:

The poles at $-1$ and $2$ distort it a bit.

This looked very close to ellipse or a degree $2$ algebraic curve to me. Tried to fit it to the form $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$.

Numerical computations suggested:

```
A,B,C,D,E,F=1,0,1.1800298285336532, -0.9962733514416349,0,-1.997900275186291
```

Plotting the algebraic curve dotted blue we get:

This appears very good approximation to me.

Q1 Is the approximation good enough?

Q2 Is it expected expression involving zeta to be well approximated by algebraic curve?

Q3 would someone try to find high precision approximation and check if $f(s)$ vanishes?