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We compute Seshadri constants $\eps(X):= \eps(\O_X(1))$ on $K3$ surfaces $X$of degrees 6 and 8. Moreover, more generally, we prove that if $X$ is anyembedded $K3$ surface of degree $2r-2 \geq 8$ in $\PP^r$ not containing lines,then $1 < \eps(X) <2$ if and only if the homogeneous ideal of $X$ is notgenerated by only quadrics (in which case $\eps(X)=3/2$).

Seshadri constants of $K3$ surfaces of degrees 6 and 8