I'd recommend:
connection: close
on pages that don't list any other resources that are stored on your server for best performance.
If you can setup a 404 error page without loading any images or special assets, then connection: close
is good.
If you need resources to completely load the error page, then connection: close
isn't as good since the connection will be closed and another one will be made to load the assets. The closing and opening of a connection increases the time for all resources to be loaded.
This kind of page works with connection: close
since no other server resources are required for the page to completely load:
<html>
<head>
<title>Not found</title>
</head>
<body>
Document is not found
</body>
</html>
This one uses resources from the local server, so I wouldn't recommend connection: close
on it.
<html>
<head>
<title>Not found</title>
</head>
<body>
<img src="http://www.your-url.com/notfound.jpg">
Document is not found
</body>
</html>
Connection: close
on such requests.