  In this paper we show for every sufficiently large integer $g$ the existence
of a complete family of closed and embedded constant mean curvature (CMC)
surfaces deforming the Lawson surfaces $\xi_{1,g}$ parametrized by their
conformal type. When specializing to the minimal case, we discover a pattern
resulting in the coefficients of the involved expansions being alternating
multiple zeta values (MZVs), which generalizes the notion of Riemann's zeta
values to multiple integer variables. This allows us to extend a new existence
proof of the Lawson surfaces $\xi_{1,g}$ for all to all $g\geq 3$ using complex
analytic methods and to give closed form expressions of their area expansion up
to order $7$. For example, the third order coefficient is
$\tfrac{9}{4}\zeta(3)$ (the first and second order term were shown to be
$\log(2)$ and $0$ respectively in \cite{HHT}). As a corollary, we obtain that
the area of $\xi_{1,g}$ is monotonically increasing in their genus $g$ for all
$g\geq 0.$
