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I would like to know more special Dirichlet L functions (like Zeta function for instace). Despite Zeta, Beta, Eta, Lambda and Hurwitz zeta are there more special Dirichlet L functions? I went to the wiki and DLMF pages on L functions and could not find any others.

Thanks

Mr. N
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    It is unclear what you mean. Do you know how to generate all the Dirichlet characters $\bmod p$ and $\bmod p^k$ thus $\bmod N$ ? Then most theorems about $\zeta(s)$ can be adapted so that the theory of Dirichlet L-functions is very similar to that of $\zeta(s)$ (except for the -conjectured not to exist- Siegel zeros part). – reuns Jan 30 '21 at 16:46
  • Sorry about the quality of the question. I edited it. I just want to know Dirichlet L functions other than the ones mentioned above. Also, if possible, their integral forms. Is it clear now? – Mr. N Jan 30 '21 at 20:12
  • @Mr.N: You may be interested by this answer where I obtain explicit formulas for Dirichlet L-functions starting with a $\zeta$ derivation (mainly for $4n\pm 1$ primes but not only...). – Raymond Manzoni Jan 31 '21 at 13:25

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$$L(s,\chi_5)=\sum_{n=1}^\infty \chi_5(n)n^{-s}=\sum_{n=0}^\infty (5n+1)^{-s}+i(5n+2)^{-s}-i(5n+3)^{-s}-(5n+4)^{-s}$$

$\chi_5$ is a character $\Bbb{Z/5Z^\times\to C^\times}$. Every Dirichlet character comes from a character $\Bbb{Z}/N\Bbb{Z^\times\to C^\times}$ for some $N$.

"their integral forms": it works mostly the same way as for $\zeta(s)$.

https://beta.lmfdb.org/Character/Dirichlet/5/2

reuns
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