On analytic continuation and functional equation of certain Dirichlet series
The well-known functional equation of the Riemann zeta function \(\zeta(s)\) is \[ 2^{1-s} \Gamma(s) \zeta(s)\cos \left( {{s\pi} \over 2} \right)= \pi^ s \zeta(1-s),\tag{1} \] where \(s\) is a complex variable. If we set \(\xi(s)=\pi^{-{s\over 2}} \Gamma({s\over 2}) \zeta(s)\), then (1) can be written as (2) \(\xi(s)=\xi(1-s)\). In this paper the authors investigate analytic continuation and functional equation of Riemann's type (see the above (1) and (2)) of Dirichlet series by using one of Riemann's methods [see \textit{E. C. Titchmarsh}, The theory of the Riemann zeta function (1986; Zbl 0601.10026)].
scientific article; zbMATH DE number 436744