1. Check to make sure you have cookies enabled in your Internet Settings.
2. You need to have TLS 1.0 & SSL 3.0 enabled, in Control Panel -> Internet Options -> Advanced tab, all the way at the bottom
3. Reset your Internet Explorer Settings
4. Make sure you're not set to "Work Offline" mode in IE.
5. Update your Internet Explorer.
6. Make sure you firewall/antivirus is allowing the lol.client.exe through
7. Try deleting the lol_air_client in C:\Riot Games\League of Legends\RADS\projects and letting it repatch
8. Try switching regions from NA to another region patch then switch back
9. Try Running the repair function in the launcher.
10. Try reinstalling
Steps: 1-5 are possible errors that can cause issues with the client, the client uses adobe air which uses internet explorers settings.
Step 6: Is simple make sure you security settings are not stoping the client from opening
Step 7: Typically fixes caused due to a bad patch.
Step 8: When doing this step see if you can open another regions client
Step 9: Reinstalling maybe faster and more likely to solve the issue but if your hesitant to reinstall try repair first note it can take upwards of 30 minutes or longer.
Step 10: Reinstall, simple a few things to do
Unistall pando media booster, league of legends and adobe air
Run a registry cleaner such as CCleaner
Make sure all previous installation files are removed
Then re install the game client.