  We continue the study of a non self-adjoint fractional three-term
Sturm-Liouville boundary value problem (with a potential term) formed by the
composition of a left Caputo and left-Riemann-Liouville fractional integral
under {\it Dirichlet type} boundary conditions. We study the existence and
asymptotic behavior of the real eigenvalues and show that for certain values of
the fractional differentiation parameter $\alpha$, $0<\alpha<1$, there is a
finite set of real eigenvalues and that, for $\alpha$ near $1/2$, there may be
none at all. As $\alpha \to 1^-$ we show that their number becomes infinite and
that the problem then approaches a standard Dirichlet Sturm-Liouville problem
with the composition of the operators becoming the operator of second order
differentiation.
