  Spectral zeta functions $\zeta(s)$ for the massless scalar fields obeying the
Dirichlet and Neumann boundary conditions on a surface of an infinite cylinder
are constructed. These functions are defined explicitly in a finite domain of
the complex plane s containing the closed interval of real axis $-1\le$ Re $s
\le 0$. Proceeding from this the spectral zeta functions for the boundary
conditions given on a circle (boundary value problem on a plane) are obtained
without any additional calculations. The Casimir energy for the relevant field
configurations is deduced.
