WordPress login issues

I managed to get a rough login form in the header of my website, https://www.le-raq.com. The problem: Login only works properly from https://www.le-raq.com/site, meaning you see the welcome message and intended links to logged in viewers. However when you login from le-raq.com, it simply clears the form.

Im guessing for some reason the form is not redirecting properly from le-raq.com. Is there any way for me to have it redirected to le-raq.com/site so it will work properly?

I can provide a test user login for you to see what im talking about.