Whilst I did notice this, I had theorised and then believed that the only reason for this was so the room opposite was 314 or pi. There are plenty of shots in the earlier seasons with this being shown and I only took this to be a little easter egg. There could however be a new reason for this number which I am looking forward to find out!
There was a mention to Cooper in the Red Room about past and future (can't recall the exact quote). This got a friend thinking about the relevance to 3 and 15 when he is with the blind lady which appears to be behind the cigarette socket. Are these numbers relating to past and future as 25 years later could put us in the year 2015 and then past could be 2003. Only a suggestion as it seems like in modern times there would easily be some police database with Dougie's picture that someone relates back to Cooper.