Focal length, I had a similar issue with my lens, not an L but with all the advice I got it came down to basics', focal length. Yeah you can up the speed a bit but try the basics' first, distance yourself a bit from the subject and take a few shots at different focal lengths with an increase to your aperature, bump that up a bit also. The issue I had with my lens was that the tip of the nose was in focus and the eyes were out of focus. I bumped up the aperature from 2.8 to 5.6 - 8 and put a few extra feet between me and what I was shooting and that fixed the problem. Then slowly adjust focal length, aperature and shot again, compare, trial and error. I'm only new to this myself but I have found going back to basics' fixes a lot of my issues, hope this is helpfull to you.