  The Bethe logarithm for a large set of states of the helium atom is
calculated with a precision of 12-14 significant digits. The numerical data is
obtained for the case of infinite mass of a nucleus. Then we study the mass
dependence and provide coefficients of the $m_e/M$ expansion, which allows us
to calculate accurate values for the Bethe logarithm for any finite mass. An
asymptotic expansion for the Rydberg states is analyzed and a high-quality
numerical approximation is found, which ensures 7-8 digit accuracy for the $S$,
$P$, and $D$ states of the helium atom.
