For what it's worth, I have had APC UPS's that would not turn on until a good battery was installed.
According to the manual for the F6C800(couldn't find the 500)
"Turn on your UPS by pressing and holding the On/Off/Test button for
approximately 3 seconds until the ON LINE Green comes on."
http://belkin.httpsvc.vitalstreamcdn.com/belkin_vitalstream_com/support/dl/unv_user_manual-a.pdf