In page 6, RH Equivalence 5.3. An equivalence of the Riemann Hypothesis says that

$$\sum_{\rho} \frac{1}{|\rho|^2} =\sum_{\rho} \frac{1}{\rho (1{-}\rho)}= 2 + \gamma - \log 4\pi$$ where $\rho$ is over nontrivial zeros of the Riemann zeta function. It's not hard to see that RH implies $$\sum_{\rho} \frac{1}{|\rho|^2}= 2 + \gamma - \log 4\pi.$$ But conversly I don't see how the equality above implies RH and there is no reference in page 6, RH Equivalence 5.3.