The L5s is based on the Gibson L5s.
This one is an early model with low impedance (dual impedance) pickups. Later they went to SuperHumbucking (Lawrence designed, three magnet or four magnet design). They also got a regular stop bar. All had the harmonica bridge. All had a matching wooden rear control plate (huge, same as W's). All had set necks. All were SOLID not hollow, and they were rather thin bodied at 1-5/8 inches like an SG. The body was solid maple. The neck was 3 piece maple (or 5 if you count the wings on the headstock).
It has been said, that the L5s was essentially a solid L5c. That is totally erroneous, except perhaps both had six strings and a neck. The L5c has pretty much standard electronics, many have only one pickup (neck). The L5c has a different tailpiece, and rosewood or ebony bridge. The binding was all different, the tuners different... the whole damn thing was different. The Gibson L5s is more of an upscale, slightly thicker L6s, due it its solid maple construction and similar (yet more refined) body shape.
What Gibson was trying to do was come out with a solid body that would appeal to the jazz player, hence the L5 name, the low impedance pickups etc... and marketed it as a luxury instrument, much as the L5c is... with gold hardware, MOP block inlay, special headstock inlay, and fancy appointments.
No slight to W, as they are due credit for making the shape available, but what you have when you build a W L5s is nothing like the Gibson L5, except the body shape.