  Let $F$ be a nonarchimedean local field of characteristic $0$ and residue
field of order not divisible by $2$. We show how to calculate the product of
the covolume of a torsion-free lattice in $PGL(2,F)$ and the formal dimension
of a discrete series representation of $GL(2,F)$. The covolume comes from a
theorem of Ihara, and the formal dimensions are contained in results of Corwin,
Moy, and Sally. By a theorem going back to Atiyah, and by triviality of the
second cohomology group of a free group, the resulting product is the von
Neumann dimension of a discrete series representation considered as a
representation of a free group factor.
