  We introduce certain lattice sums associated with hyperplane arrangements,
which are (multiple) sums running over integers, and can be regarded as
generalizations of certain linear combinations of zeta-functions of root
systems. We also introduce generating functions of special values of those
lattice sums, and study their properties by virtue of the theory of convex
polytopes. Consequently we evaluate special values of those lattice sums,
especially certain special values of zeta-functions of root systems and their
affine analogues. In some special cases it is possible to treat sums running
over positive integers, which may be regarded as zeta-functions associated with
hyperplane arrangements.
