  In 2009, H. Akatsuka introduced the \emph{zeta Mahler function} (ZMF, also
called \emph{zeta Mahler measure}) related to the \emph{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.
