I ran it from the terminal and there were no errors (and no output other than the starting messages on boot).
emily@ubu:~$ /usr/share/playonlinux/playonlinux --run "League of Legends" %F
Looking for python... 2.7.6 - selected
[main] Message: PlayOnLinux (4.2.6) is starting
[clean_tmp] Message: Cleaning temp directory
Script started /home/emily/.PlayOnLinux/shortcuts/League of Legends
[POL_System_CheckFS] Message: Checking filesystem for lol.launcher.admin.exe
[POL_Wine] Message: Running wine-1.7.33-LeagueOfLegends2 lol.launcher.admin.exe %F (Working directory : /home/emily/.PlayOnLinux/wineprefix/LeagueOfLegends/drive_c/Riot Games/League of Legends)
[POL_Wine] Message: Notice: PlayOnLinux deliberately disables winemenubuilder. See http://www.playonlinux.com/fr/page-26-Winemenubuilder.html
[POL_Wine] Message: Wine return: 0
Ubuntu 14.04 64-bit, fresh install (as of 2 days ago)
PlayOnLinux 4.2.6
CPU: Xeon E3-1200
Graphics: GeForce GTX 660 Ti
RAM: Corsair Vengence LP 2 x 8 GB, 1600 (running at 1333)
Let me know if there's anything else specific that would be useful to know (and maybe how to find it out).