  The class of Dirichlet series associated with a periodic arithmetical
function $f$ includes the Riemann zeta-function as well as Dirichlet
$L$-functions to residue class characters. We study the value-distribution of
these Dirichlet series $L(s;f)$, resp. their analytic continuation in the
neighborhood of the critical line (which is the abscissa of symmetry of the
related Riemann-type functional equation). In particular, for a fixed complex
number $a\neq 0$, we prove for an even or odd periodic $f$ the number of
$a$-points of the $\Delta$-factor of the functional equation, prove the
existence of the mean-value of the values of $L(s;f)$ taken at these points,
show that the ordinates of these $a$-points are uniformly distributed modulo
one and apply this to show a discrete universality theorem.
