  The first two papers in this series prove the Harris-Venkatesh conjecture and
its refinement with the Stark conjecture for imaginary dihedral modular forms
of weight $1$. This paper explicitly describes the constants appearing in the
Harris-Venkatesh (plus Stark) conjecture for dihedral modular forms by
evaluating $\mathrm{GL}(2) \times \mathrm{GL}(2)$ Rankin--Selberg periods and
zeta integrals on newforms and optimal forms. One consequence is a formula for
the ratio between Petersson norms and adjoint $L$-values. Our calculations also
extend to exotic modular forms whose level is odd or whose Deligne-Serre
representation is $2$-ordinary.
