I have not had this issue before to the extent where the line would not show up while the computer was off, but the computer is on. As a point in the possible correct direction, I would check the graphics card drivers to make sure that they are up to date, if they are, re install them. If you have custom graphics card software, check to make sure there aren't any weird settings checked. I don't know why it would help, but for some reason I think it would be a good idea to try hitting the auto-adjust button on the monitor to see if that fixes it.
Below are two more questions for you.
When did this problem start, or has it always been there?
Did something change right before the problem started (Windows Updates, new graphics card, etc)?