1. Do you have any plugins installed? If so deactivate them and see if it helps.
2. Revert to Twenty Twelve theme and see if it helps.
3. Check your content/uploads folder and look for any wrongly placed files.
I have been digging a little more, and the breakdown of the call in tools.pingdom.com shows the origins as a WordPress codec file in the root: xmlrpc.php
WordPress lists this file as being enabled as standard since 3.5: With WordPress XML-RPC support, you can post to your WordPress blog using many popular Weblog Clients. The XML-RPC system can be extended by WordPress Plugins to modify its behavior.
I’ve tried to rename the file but the bad request still remains. I will work through your 1-3 advice above, as perhaps a plugin is in conflict with WordPress.
I tried deactivating the plugins and switching theme to Twenty Twelve did not make a difference, though I would not feel to confident about examining the content/uploads folders to see if anything there is incorrectly placed. Also downloaded a plugin to disable XML-RPC, but the bad call is still active, and slowing the site load by 2-3 seconds.