The integral in question is:
![\int_{1/2}^{\infty} \left[\zeta\left(t+i\mathrm{Im}(\rho_{n})\right)-1\right] dt](https://s0.wp.com/latex.php?latex=%5Cint_%7B1%2F2%7D%5E%7B%5Cinfty%7D+%5Cleft%5B%5Czeta%5Cleft%28t%2Bi%5Cmathrm%7BIm%7D%28%5Crho_%7Bn%7D%29%5Cright%29-1%5Cright%5D+dt&bg=000000&fg=b8babb&s=0 "\int_{1/2}^{\infty} \left[\zeta\left(t+i\mathrm{Im}(\rho_{n})\right)-1\right] dt")
Since the limit starting from a nontrivial root going parallel to the real axis of the zeta function is 1, we may subtract that off so that it converges.
144 evaluations, text file, done with mpmath
What’s amazing, is that when the values of this integral corresponding to nontrivial roots of the zeta function are plotted, some interesting structure revealed:
