I would like to test whether or not the following product (or its complement)

$$\displaystyle \displaystyle \prod_{n=1}^\infty \left(1- \frac{s}{\frac12+ (-1)^n\, \gamma_n \, i} \right)$$

converges to fractions of known constants for e.g. $s=2n$. Note that $\gamma_n$ is the imaginary part of the n-th non-trivial zero of $\zeta(s)$.

I am aware of Andrew Odlyzko's tables of non-trivial zeros and using those I could calculate the product up to $n=2,001,052$. However convergence of the product is very slow and I am keen to get access to a table with couple of million zeros more (with $\gamma_n$ at 9 digits).

Grateful for any hints/tips on where I might to find these and/or for a fast tool/program (e.g. using the Riemann-Siegel formula) to generate them.



I found the zeros I needed and post this as an answer rather than an update.

There is a very nice website that gives access to a database with 'gazillions' of non-trivial zeros (with even higher accuracy than the Odlyzko tables). The only issue is that it limits downloads to chunks of 100000 max. I managed to easily download the first 5mln zeros in 50 .tex files of 100000 zeros in less than 30 minutes.

The steps are easy:

  1. Enter in List = 100000 and the relevant starting value. Press Go.
  2. A HTML screen pops up with all imaginary parts in text format.
  3. Wait a few seconds for the page to fully load and then right click on the screen.
  4. Press 'Save Page As' and type the filename. Choose Format = 'Page Source'. Press 'Save'.

This solves my problem, however I did not find a nice relation to known constants (actually the product is complex). The only 'interesting' fact I found, is that the absolute value of the 'flipping' product of zeros is the square root of the Riemann $\xi(s)$ function.


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