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.

10$\begingroup$ see store.doverpublications.com/0486417409.html especially reproduction on page 156 and discussion of Riemann's hand calculations from about 155162 $\endgroup$ – Will Jagy Sep 19 '13 at 4:32
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).
The method he used was the expansion that is now called the RiemannSiegel
formula. I did not see any use, e.g., of an approach based on
EulerMaclaurin. The limited accuracy Riemann obtained reflects that of
the error term in the RS formula.
"Know" is hard for those of us without a ouija board, but I think people believe that the RiemannSiegel formula was used.

7$\begingroup$ I'm looking at Edward's book, this is discussed in pages 155162 and on. He reproduces the page with the formula from the Gottingen Library. $\endgroup$ – Will Jagy Sep 19 '13 at 3:06