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

1.

2.

3.

s = 1/2

Trivial.

s = 1/√2 = .707+

Trivial.

s = (1+√7) / 4 = .911+

Found by Erich Friedman

in July @2011.

4.

5.-6.

7.

s = (√6 + √2) / 4 = .965+

Found by Maurizio Morandi

in July 2012.
s = 1

Trivial.

s = (2 + √2) / 3 = 1.138+

Found by David W. Cantrell

in July 2012.

8.

9.

10.

s = (1 + 2√2) / 3 = 1.276+

Found by Maurizio Morandi

in July 2012.
s = 4/3 = 1.333+

Found by Maurizio Morandi

in July 2012.
s = 1.400+

Found by Maurizio Morandi

in July 2012.

11.-12.

13.

14.

s = √2 = 1.414+

Trivial.

s = 1.499+

Found by Maurizio Morandi

in July 2012.
s = 1.607+

Found by Maurizio Morandi

in August 2012.

15.

16.

17.

s = 1.680+

Found by Maurizio Morandi

in August 2012.
s = 1/3 + √2 = 1.747+

Found by Maurizio Morandi

in August 2012.
s = (4 + √2) / 3 = 1.804+

Found by Maurizio Morandi

in August 2012.