In a recent paper, S. Chatterjee determined the leading order term of the free energy of U(N) lattice Yang-Mills theory in $\Lambda_n=\{0,\ldots,n\}^d\subset \bZ^d$, for every $N\geq 1$ and $d\geq 2$. The formula is explicit apart from a contribution $K_d$ which corresponds to the limiting free energy of lattice Maxwell theory with boundary conditions induced by the axial gauge. After a brief motivation, I recall some of the key steps to obtain the leading order term of the free energy and I explain an equivalent characterization of $K_d$ that admits its explicit computation, for every $d\geq 2$.