The famous plot of $\zeta(1/2+it)$ for real $t$ seems to show that this function gets a non zero real value exactly once between two consecutive Riemann zeros. Moreover, letting $\rho\_{i}$ the $i$-th critical zero of positive imaginary part, it seems that such a non zero real value is attained at around $1/2(\rho\_{i}+\rho\_{i+1})$.


My question is: are there rigorous proofs for this phenomenon? 



Is the number of real values of Zeta on the critical line up to some ordinate known?
$\zeta(1/2+i g\_n)$ is nonzero real iff $g\_n$ if [Gram point](https://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_theta_function#Gram_points), which is efficienly computable and you can use binary search to count up to given $t$.


Your claim "his function gets a non zero real value exactly once between two consecutive Riemann zeros" is [Gram's law](https://en.wikipedia.org/wiki/Riemann_hypothesis#Gram_points), which is known to be false.


