May be I am missing something. But Plot can plot Piecewise as is.
L3 = Piecewise[{{1, x <= 0}, {4, 0 < x <= 1}, {5, 1 < x <= 3}, {2, 3 < x <= 5}}, 1];
Plot[L3, {x, -3, 4}, Exclusions -> None]
btw, it is not really a good idea to use the single letter l as a variable, since on most fonts and on the screen it looks like 1 and makes it hard to read the code. In this case, using L1 or L2 is ok, even though we broke the golden rule of using UpperCase for first letter for a variable, but since it is a 2 letter variable, and has a digit in the sescond place, there is no chance that WL will use such a symbol in the future. So it is safe to use it and it makes the code easier to read.
