There are two options for solve this problem, you need to switch your connection with your DNS or Google DNS
Google DNS are: 8.8.8.8 and below 4.4.4.4
Check the video for fix it!
if the metod in the video not work you can try to do this:
1. Close out your game/patcher, hit ctrl+alt+delete and make sure all instances of LoL.Launcher are gone, then restart the patcher process and let it sit.
2. Go to your Riot Games/RADS/System folder under your default install path, right click and run the rads_user_kernel.exe file as an admin, then try starting your patcher.
3. Try deleting your LoL_Air_Client folder the default install path for this folder is c:\Riot Games\RADS\Projects and then repatching.
3. See this article, About changing to a public DNS
4. Replace installdirectory/RADS/system/rads_user_kernel.exe, with this file
5. Try installing, Microsoft Dot NET Framework 4.0 and, Dot NET 3.5
6. Try running through the RADS solution in this Video Fix
Follow me on Youtube for more video