  Let $D$ be a diagram of an alternating knot with unknotting number one. The
branched double cover of $S^3$ branched over $D$ is an L-space obtained by half
integral surgery on a knot $K_D$. We denote the set of all such knots $K_D$ by
$\mathcal D$. We characterize when $K_D\in \mathcal D$ is a torus knot, a
satellite knot or a hyperbolic knot. In a different direction, we show that for
a given $n>0$, there are only finitely many L-space knots in $\mathcal D$ with
genus less than $n$.
