  Greene, Lewallen and Vafaee characterized $(1,1)$ L-space knots in $S^3$ and
lens space in the notation of coherent reduced $(1,1)$-diagrams. We analyze
these diagrams, and deduce an explicit description of these knots. With the new
description, we prove that any L-space obtained by Dehn surgery on a
$(1,1)$-knot in $S^3$ has non-left-orderable fundamental group.
