This paper focuses on the characterization of the stability crossing curves of a class of linear systems with shifted fractional gamma-distributed delay. First, we describe the frequency crossing set, i.e., the set of frequencies where the characteristic roots may cross the imaginary axis as the parameters change. Next, we describe the stability crossing curves, i.e., the set of parameters (average delay, gap) such that there exists at least one characteristic root on the imaginary axis. Such stability crossing curves divide the parameter-space into different regions, such that within each such region, the number of strictly unstable roots is fixed. The classification of the stability crossing curves is also discussed and illustrated.