  Building on the mapping relations between analytic functions and periodic
functions using the abstract operators $\cos(h\partial_x)$ and
$\sin(h\partial_x)$, and by defining the Zeta and related functions including
the Hurwitz Zeta function and the Dirichlet L-function in the form of abstract
operators, we have obtained many new series expansions associated with these
functions on the whole complex plane, and investigate the number theoretical
properties of them, including some new rapidly converging series for
$\eta(2n+1)$ and $\zeta(2n+1)$. For $n\in\mathbb{N}$, each of these series
representing $\zeta(2n+1)$ converges remarkably rapidly with its general term
having the order estimate: \[O(m^{-2k}\cdot
k^{-2n+1})\qquad(k\rightarrow\infty;\quad m=3,4,6).\]
