In this paper we study for a given azimuthal quantum number the eigenvalues of the Chandrasekhar-Page angular equation with respect to the parameters : and : , where is the angular momentum per unit mass of a black hole, is the rest mass of the Dirac particle and is the energy of the particle (as measured at infinity). For this purpose, a self-adjoint holomorphic operator family associated to this eigenvalue problem is considered. At first we prove that for fixed the spectrum of is discrete and that its eigenvalues depend analytically on . Moreover, it will be shown that the eigenvalues satisfy a first order partial differential equation with respect to and , whose characteristic equations can be reduced to a Painlevé III equation. In addition, we derive a power series expansion for the eigenvalues in terms of and , and we give a recurrence relation for their coefficients. Further, it will be proved that for fixed the eigenvalues of are the zeros of a holomorphic function which is defined by a relatively simple limit formula. Finally, we discuss the problem if there exists a closed expression for the eigenvalues of the Chandrasekhar--Page angular equation.