Hi Ten,
You can test this fairly easily with curl:
curl -v http://1.1.1.1/index.html
LTM will set a persistence cookie any time a load balancing decision has been made regardless of how many pool members are up.
Is the authentication URL on the same fully qualified domain name? If not, is it on the same domain? Does the authentication URL go to the same virtual server?
Do you have a OneConnect profile enabled on the virtual server? If you're using SNAT you can use the default OneConnect profile with a /0 source mask. If you're not using SNAT then configure a custom OneConnect profile with a /32 source mask. See this OneConnect wiki page for details:
http://devcentral.f5.com/wiki/default.aspx/AdvDesignConfig/oneconnect.html
Aaron