A lattice Green function (LGF) of the \(d\)-dimensional face-centered cubic lattice is given by a \(d\)-fold integral whose expansion around the origin is hard to obtain as the dimension goes higher. There is a recursive method to generate the expansion of LGF. In the paper, the authors show the strength and the limit of the method by producing some series and corresponding linear differential equations for \(d=8,9,\dots,12\).