  The analytic result for the singlet part of the Adler function of the vector
current in a general gauge theory is presented in five-loop approximation.
Comparing this result with the corresponding singlet part of the
Gross-Llewellyn Smith sum rule [1], we successfully demonstrate the validity of
the generalized Crewther relation for the singlet part. This provides a
non-trivial test of both our calculations and the generalized Crewther
relation. Combining the result with the already available non-singlet part of
the Adler function [2,3] we arrive at the complete ${\cal O}(\alpha_s^4)$
expression for the Adler function and, as a direct consequence, at the complete
${\cal O}(\alpha_s^4)$ correction to the $e^+ e^-$ annihilation into hadrons in
a general gauge theory.
