Explicit Formula for $\zeta(s)$

Question

In the explicit expression for $$\psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2} \log (1-x^{-2}) $$ $ x^\rho$ denotes $x^{\mathrm{Re} \rho}$. I wanted to know if there is some formula for $$\sum_{\rho} \frac{x^{\rho}}{\rho}$$ and whether that divisor also denotes the real part of $\rho$ or not.