Does jailbreakme.com work for 3GS 3.1.3?

[NHL Fever]

I have an Iphone 3GS 3.1.3, new bootrom. Never before jailbroken, no SHSH files saved. I remain very confused on whether jailbreakme.com works for my situation. On a number of sites it lists many versions that it works for, including mine, but then the instructions start off by saying to update to 4.0.0 or 4.0.1. Many people like myself cannot restore to those versions, having no SHSH files saved for them. So what is it? Does it work for 3.1.3 itself?, or do they say that only because people on 3.1.3 are supposed to be able to restore to 4.0.1 and then have it work?

Any responses greatly appreciated

[j3st3r]

yes it will jailbreak 3.1.3, you can also use spirit to jailbreak, its your choice

[Sailor-girl]

I'm jailbroken with spirit on 3.1.3 new bootrom. I want to go to ios4 only if it's jailbreakable. What should I do?

I have shsh blobs saved in cydia for 3.1.3, 4.0 and 4.01 if that makes a difference.