Error code 50x are caused by external application and are not webserver related, so you should find error_log in the directory where the error happened if there is no path defined.
FYI: LiteSpeed can be configured to Auto Fix 503 Errors in WebAdmin Console.
Specifies whether to try to fix the "503 Service Unavailable" error by restarting the server gracefully. A "503" error is usually caused by malfunctioning external applications and a web server restart can often fix the error temporarily. If enabled, the server will restart automatically whenever there are more than 30 "503" errors within a 30 seconds span.
I hope that helps?