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# L-functions with Riemann's functional equation and the Riemann hypothesis

## ##article.authors##

• Nakamura, Takashi Department of Liberal Arts, Faculty of Science and Technology, Tokyo University of Science

## キーワード:

L-functions、 Riemann's functional equation、 Riemann hypothesis

## 抄録

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.

## 利益相反に関する開示

The authors have no conflicts of interest directly relevant to the content of this article.

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## 引用文献

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## 改版理由

Remarks on infinite product representations are added.