There are two options for solve this problem, you need to switch your connection with your DNS or Google DNS
Google DNS are: 126.96.36.199 and below 188.8.131.52
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
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