If cookie is injected by the F5 and we lose an http server, will those sessions cookies have to be re-established

If the client presents a cookie that points at a server which is no longer available, the BIG-IP will make a load-balancing decision and send the client connection to a different back-end server. When the BIG-IP updates the cookie, the cookie will contain information identifying the new back-end server as the destination to persist to.