  Using the analytical $\rm{\overline{MS}}$-scheme three-loop contribution to
the perturbative Coulomb-like part of the static color potential of heavy
quark-antiquark system, we obtain the analytical expression for the
fourth-order $\beta$-function in the gauge-invariant effective V-scheme in the
case of the generic simple gauge group. Also we present the Adler function of
electron-positron annihilation into hadrons and the coefficient function of the
Bjorken polarized sum rule in the V-scheme up to $\alpha^4_s$ terms. We
demonstrate that at this level of PT in this effective scheme the
$\beta$-function is factorized in the conformal symmetry breaking term of the
generalized Crewther relation, which connects the flavor non-singlet
contributions to the Adler and Bjorken polarized sum rule functions. We prove
why this relation will be true in other gauge-invariant renormalization schemes
as well. The obtained results enable to reveal the difference between the
V-scheme $\beta$-function in QED and the Gell-Man--Low $\Psi$-function. This
distinction arises due to the presence of the light-by-light type scattering
corrections first appearing in the static potential at the three-loop level.
