  In 1969, I. Bernstein and S. Gelfand introduced an object, which is now
called the zeta Mahler function (ZMF, also zeta Mahler measure) and related to
the Mahler measure.
  Here we discuss a family of ZMFs attached to the Laurent polynomials $k +
(x_1 + x_1^{-1}) \cdots (x_r + x_r^{-1})$, where $k$ is real. We give explicit
formulae, present examples and establish properties for these ZMFs, such as an
RH-type phenomenon. Further, we explore relations with the Mahler measure.
