Does anyone know how Riemann calculated the first few non-trivial 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.



How did Riemann calculate the first few non-trivial zeros of the zeta-function?
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 Riemann-Siegel
formula. I did not see any use, e.g., of an approach based on
Euler-Maclaurin. The limited accuracy Riemann obtained reflects that of
the error term in the R-S formula.


