Heavily based on Agno's question.

Define:

$$ \chi(s)=\pi^{-(\frac{s}{2})} \Gamma(\frac{s}{2}) $$

Agno conjectured: only for $\sigma=\frac12$, $\Re(\chi(s)) = \Re(\zeta(s)) =0$ is always true, except when $s=\rho_n$ (assuming RH).

This turned out to be false.

Limited numerical evidence suggests that this might be true on the critical line, which would give alternative way to find vanishing of $\Re\zeta(1/2+it)$ except at zeros and maybe some sort of closed form.

This appears non-trivial result to me.

Q1. Is it true that $\Re\chi(1/2+it)=0 \implies \Re\zeta(1/2+it)=0$ except at zeros?

Q2. Is it true for the first $4$ vanishing of $\Re\zeta(1/2+it)$ except at zeros?

According to sage:

$$ \Re\chi(1/2+it) = \frac{\cos\left(-\frac{1}{2} \, t \log\left(\pi\right)\right) \Re \left( \Gamma\left(\frac{1}{2} i \, t + \frac{1}{4}\right) \right)}{\pi^{\frac{1}{4}}} - \frac{\Im \left( \Gamma\left(\frac{1}{2} i \, t + \frac{1}{4}\right) \right) \sin\left(-\frac{1}{2} \, t \log\left(\pi\right)\right)}{\pi^{\frac{1}{4}}} \qquad (1) $$

Q3. Can (1) be simplified further?

Plot: