I believe there is a somewhat simpler way to determine that the L-blocks must all contain only odd numbers.

If one looks at any row (or column) of 2x2 blocks, there are 6 blocks each containing 4 values and 2 L-blocks each containing 3 values (when the 10s are excluded). In these blocks we need to place 10 odds, 8 evens and 12 tens with the restriction that each block can only contain values of the same type. The only way the odds can be accommodated is by using one 4-block and both L-blocks. The remaining groups fit comfortably in 2 and 3 4-blocks respectively.

I still can't understand how anyone can solve this puzzle in 41 minutes! :)