  We present an abstract framework to study weak convergence of numerical
approximations of linear stochastic partial differential equations driven by
additive L\'evy noise. We first derive a representation formula for the error
which we then apply to study space-time discretizations of the stochastic heat
equation, a Volterra-type integro-differential equation, and the wave equation
as examples. For twice continuously differentiable test functions with bounded
second derivative (with an additional condition on the second derivative for
the wave equation) the weak rate of convergence is found to be twice the strong
rate. The results extend earlier work by two of the authors as we consider
general square-integrable infinite-dimensional L\'evy processes and do not
require boundedness of the test functions and their first derivative.
Furthermore, the present framework is applicable to both hyperbolic and
parabolic equations, and even to stochastic Volterra integro-differential
equations.
