Let $\{X_{i}\}$ be a sequence of independent 
identically distributed random variables with
$EX_{1}>0$, and let $\{S_{k}\}$ be the sequence 
of the partial sums.
We obtain asymptotic expansions for the renewal measure
$\sum_{k=0}^{\infty}P(S_{k}\in \cdot)$,
taking into account the influence of the roots
of the characteristic equation $1-E\exp(sX_{1})=0$
which lie in the strip of analyticity of 
the Laplace transform $E\exp(sX_{1})$.
The exact asymptotic behaviour of the remainder terms 
is established.
We also give submultiplicative estimates for the 
remainders.