  An ACM bundle on a polarized algebraic variety is defined as a vector bundle
whose intermediate cohomology vanishes. We are interested in ACM bundles of
rank one with respect to a very ample line bundle on a K3 surface. In this
paper, we give a necessary and sufficient condition for a non-trivial line
bundle $\mathcal{O}_X(D)$ on $X$ with $|D|=\emptyset$ and $D^2\geq L^2-6$ to be
an ACM and initialized line bundle with respect to $L$, for a given K3 surface
$X$ and a very ample line bundle $L$ on $X$.
