For every g >>1, we show the existence of a complete and smooth family of closed constant mean curvature surfaces f^g_φ , φ ∈ [0, π/2 ], in the round 3-sphere deforming the Lawson surface ξ_{1,g} to a doubly covered geodesic 2-sphere with monotonically increasing Willmore energy. To construct these we use an implicit function theorem argument in the parameter t =1/(2g+2).This allows us to give an iterative algorithm to compute the power series expansion of the DPW potential and area of f^g_φ at t = 0 explicitly. In particular, we obtain for large genus Lawson surfaces ξ_{1,g} a scheme to explicitly compute the coefficients of the power series in t in terms of multiple polylogarithms. Remarkably, the third order coefficient of the area expansion is identified with 9/4 ζ(3), where ζ is the Riemann ζ function (while the first and second order term were shown to be log(2) and 0 respectively in [12]).