If you can get the phone to backup cleanly, or have a backup from the upgrade, you should probably go for a DFU update -
Pop the phone into DFU mode and get iTunes to reflash it from scratch.
I've worked in mobile software for 14 years, and it's my opinion that updating mobile devices over the air with partial updates is a bad idea. I always update and migrate iPhones using iTunes and a cable. It's the best way to be safe, from the start.
When your iPhone tells you to update, do it with iTunes. Pure and simple. If you're not near your PC or mac, wait until you are.
I have tried this and I have again lost my phone. It remains black and the drivers install again then it goes to preparing phone for restore - the apple logo appears with the bar and then itunes just says waiting for iphone for ages and then it comes up with the iphone logo and connector sign. It then says
"iphone could not be restored an unknown error occured 4013"
And now nothing on my iphone again!
There's another couple of answers on that iTunes error -
https://discussions.apple.com/thread/5320142?start=15&tstart=0 - there's a couple of successes here
https://discussions.apple.com/message/23422298#23422298 - security software issues on the PC?
Otherwise take a look through some more via google -
The consensus on fixing seems to revolve around the PC running iTunes rather than the iPhone in what I see here. You may need to uninstall iTunes and reinstall it, or try using someone else's iTunes to do the software restore.
Good luck and check the solutions in the links above.