I don't know if this helps but you can put $D=24$ in (13) (14) of my paper to get an explicit formula for $\tau(n)$.
MR2218820 (2007c:17009) Westbury, Bruce W. Universal characters from the Macdonald identities. Adv. Math. 202 (2006), no. 1, 50--63.
doi:10.1016/j.aim.2005.03.013
Since $SL(5)$ is a simple Lie algebra of dimension 24 this also relates $\tau(n)$ to the affine root system of type $A_4$.
I doubt Lehmer would have had this in mind.