We introduce and present the general solution of non-self-adjoint fractional differential equations formed by the composition of a left Caputo and left Riemann– Liouville fractional integral under 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 a, 0<a < 1, there is a finite set of real eigenvalues and that, for a; near 1/2, there may be none at all. As a approaches 1 from the left , 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. Finally, two sided estimates as to their location are provided as is their asymptotic behavior as a function of a.