  We study the effects of non-magnetic impurity vertex correction on nuclear
spin-lattice relaxation rate $1/{T_1} $ of conventional s-wave superconductors
within the Eliashberg formalism. We obtain, with a self-consistent $t-$matrix
treatment of impurity scatterings, the expressions for impurity vertex function
and nuclear spin-lattice relaxation rate. The $1/T_1$ is evaluated with a
simple approximation on angular average, and found to agree in clean limit with
the previous result that $1/(T_1 T)$ remains unrenormalized under the impurity
vertex correction. As dirtiness is increased, on the other hand, the coherence
peak in $1/(T_1 T)$ is found to increase due to the impurity vertex correction.
This result is discussed in connection with the experimental observations on
conventional superconductors.
