The following pictures show L's (with short side 1) packed inside the smallest known L (with short side s).

1.

2.-4.

5.

s = 1

Trivial.

s = 2

Trivial.

s = 5/2 = 2.5

Found by Maurizio Morandi

in July 2012.

6.

7.-9.

10.

s = 1 + 4√2/3 = 2.885+

Found by Maurizio Morandi

in July 2012.
s = 3

Trivial.

s = 7/2 = 3.5

Found by Maurizio Morandi

in July 2012.

11.

12.

13.-16.

s = 7/3 + √2 = 3.747+

Found by Maurizio Morandi

in July 2012.
s = (19 + 9√2) / 8 = 3.965+

Found by Maurizio Morandi

in July 2012.
s = 4

Trivial.