Going to look into this game, sorry for the inconvenience.
(btw, the 'try checking your internet connection' message doesn't mean you DON'T have internet, it means that is the first thing you should confirm if you are trying to troubleshoot the problem)
EDIT: Just looked at some of the server logs and there is something funky going on. It would be super useful if you could get me your client r3d logs so we can have a look at what was going on there. Here are steps to do that:
Most recent R3d log. Here are the steps to do that:
a. Navigate to the folder where you have League installed on your computer. For me it is:
Local Disk (C:)/Riot Games/League of Legends
b. Navigate to Logs
c. Navigate to GameLogs. For me it looks like this:
Local Disk (C:)/Riot Games/League of Legends/Logs/GameLogs
d. Open up the most recent folder (you can sort by date modified to find the most recent one), and open the text document that ends in 'r3dlog'
e. Copy the text document to your clipboard (hotkey shortcut is press ctrl+a and then ctrl+c
f. Paste this to pastebin. Go to https://pastebin.com. Click in the text box just below ‘New Paste’ and press ctrl+v
g. Click the ‘Create New Paste’ button towards the bottom left of the screen , and then copy the URL from your browser. Here is what an example url will look like:
https://pastebin.com/t1t93uZK
h. Provide that URL in your post