Dirichlet L functions

Question

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

Answer 1

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

$\chi_5$ 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