The presence of trivial zeros is required by the functional equation. In its analytic normalization, the L-function $L(s)$ has no zeros or poles in the right half-plane $\Re(s)>1$, but the gamma factors that appear in the functional equation have poles. Since the completed L-function has no poles (except possibly from Riemann zeta-function factors), the L-function must have zeros at poles of the gamma factors.
For arithmetic L-functions, the poles are at certain negative integers.
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- Last edited by David Farmer on 2019-05-14 07:36:16
- 2019-05-14 07:36:16 by David Farmer
- 2019-05-14 06:51:07 by David Farmer
- 2019-05-14 06:38:03 by Andrew Sutherland