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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.

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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.

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

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    $\begingroup$ I'm looking at Edward's book, this is discussed in pages 155-162 and on. He reproduces the page with the formula from the Gottingen Library. $\endgroup$ – Will Jagy Sep 19 '13 at 3:06

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