One of the common issues with the much appreciated L2 G2 phone is that the mobile data does not work at times. Some people find that even when they have full signal, they are not able to connect to mobile data.
The workaround for this is turning the airplane mode on and off. You can do this by holding down the Power Key, until you see the Turn-On Airplay Mode option, tap it and then press OK. Another way of getting to the airplane mode is by going to apps- settings- tethering and network- airplane mode.
A software update has been initiated from LG which should solve the problem, when you are connected to the WI-FI network. You can also check for the update by going to Menu- All Settings – General- About Phone- Software Update- Update Now