The spectral measures for self-adjoint elements in $\mathbb C F_2$ are very special. In particular, it is known that non of the elements in $\mathbb C F_2$ has a kernel when acting via the left-regular representation on $\ell^2 F_2$. This was shown by Peter Linnell using index-theoretic methods in
Linnell, Peter, Division rings and group von Neumann algebras. Forum Math. 5 (1993), no. 6, 561–576.
(It also follows from another of Linnell's papers and the fact that $F_2$ is left-orderable.)
It is also known that Novikov-Shubin invariants are always positive. This together implies that $\Delta(T) \neq 0$ of if $T\neq 0$.
The main advantage is that $\sum_{n} \tau(T^n)z^n$ is an algebraic power series for any $T \in \mathbb C F_2$. This implies the result about Novikov-Shubin invariants (and much more). It was proved in
Sauer, Roman, Power series over the group ring of a free group and applications to Novikov-Shubin invariants. High-dimensional manifold topology, 449–468, World Sci. Publ., River Edge, NJ, 2003
Starting with the computations in this paper (and using the Stieltjes-Inversion formula), you can also make concrete computations for the determinant of specific elements.
