L-functions with Riemann's functional equation and the Riemann hypothesis
DOI:
https://doi.org/10.51094/jxiv.238Keywords:
L-functions, Riemann's functional equation, Riemann hypothesisAbstract
Let $\chi_4$ be the non-principal Dirichlet character mod $4$ and $L(s,\chi_4)$ be the Dirichlet $L$-function associated with $\chi_4$, and put $R(s):= s 4^{s} L(s+1,\chi_4) + \pi L(s-1,\chi_4)$. In the present paper, we show that the function $R(s)$ has the Riemann's functional equation and its zeros only at the negative even integers and complex numbers with real part $1/2$. We also give other $L$-functions that have the same property.
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Submitted: 2023-01-03 23:58:50 UTC
Published: 2023-01-05 06:52:57 UTC — Updated on 2023-02-01 04:40:52 UTC
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- 2023-02-01 04:40:52 UTC (3)
- 2023-01-16 02:55:32 UTC (2)
- 2023-01-05 06:52:57 UTC (1)
Reason(s) for revision
Remarks on infinite product representations are added.License
Copyright (c) 2023
Takashi Nakamura
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