First of all you need the ThomasV key.
Execute the command line on terminal:
gpg --keyserver keyserver.ubuntu.com --recv 6694D8DE7BE8EE5631BED9502BD5824B7F9470E6
Go to Electrum website and download Electrum-3.1.3.tar.gz and its signature Electrum-3.1.3.tar.gz.asc
Copy both files to the same folder, open the terminal and use command 'cd' to navigate to that folder and run the command line:
gpg --verify Electrum-3.1.3.tar.gz.asc Electrum-3.1.3.tar.gz
If the message returned says Good signature and that it was signed by ThomasV with a pubkey 7F9470E6, then the software is authentic.