@KConrad Sorry for my poorly formed question. Thank you so much.

@KConrad et al, do we know $N(t)$ for $L(s,\chi_4)$? Thanks

@Stopple The proof in Edwards 'Riemann Zeta Function' on page 176, says "if the Riemann hypothesis is true, this derivative is not only positive (because all terms are positive) but also very large [because by von Mangoldt's estimate of N(t) the 𝛼's must be quite dense] between successive zeros of Z". For having the derivative positive all we need are 𝛼−𝑡 being real. I don't think "also very large" part is necessary for the proof. It would be good to get clarity. I am only interested in $t$ real. Edwards needed RH to be true to make $\alpha - t$ terms positive (he needed $t$ real).

@Stopple Naive question. Why do we have to assume RH to be true? Aren't all the zeros of $\zeta(0.5 + it)$ on the critical line?

Good point. You think periodicity is the property that we need? If the sequence is periodic, then JI may hold true.