Does anyone know how Riemann calculated the first few nontrivial zeros of the Zeta function? I am wondering if he approximated the integral, $\frac{1}{2 \pi i} \int_{R} \frac{{\xi}^\prime(z)}{\xi (z)} dz$ over appropriate rectangle(s) in the critical strip. This still seems difficult, however, without a computer.

In searching through the Riemann Nachlass in Gottingen (including those
folders not listed as connected with \zeta(s)) there is no
evidence  at least that has been saved  that Riemann computed
anything more than the first few zeros (I think up to ordinate about 80). 


"Know" is hard for those of us without a ouija board, but I think people believe that the RiemannSiegel formula was used. 

