  The location of zeros of the basic double sum over the square lattice is
studied. This sum can be represented in terms of the product of the Riemann
zeta function and the Dirichlet beta function, so that the assertion that all
its non-trivial zeros lie on the critical line is a particular case of the
Generalised Riemann Hypothesis (GRH). The treatment given here is an extension
of that in two previous papers (arxiv:1601.01724, 1602.06330), where it was
shown that non trivial zeros of the double sum either lie on the critical line
or on lines of unit modulus of an analytic function intersecting the critical
line. The extension enables more specific conclusions to be drawn about the
arrangement of zeros of the double sum on the critical line, which are
interleaved with zeros of analytic functions, all of which lie on the critical
line. Possible arrangements of zeros are studied, and it is shown that in all
identified cases the GRH holds.
