iPhone 5 problem, "No Service" and not being recognised by iTunes

I've recently purchased a second hand iPhone 5. They demonstrated the phone making calls before I purchased it.

Unfortunately the phone isn't being recognised by iTunes, and is also very frequently showing me the message "No service". I did however very briefly have service earlier today, enough to send and receive a couple of SMS's.

The things I have tried are:

• Restarting laptop,


• Trying another laptop,


• Uninstalling and reinstalling iTunes (on both laptops),


• "Reset Network Settings,


• Rebooting phone,


• Airplane mode on / off,


• 4G on / off,


• Updating all software,


• Trying to turn off automatic carrier selection, but I get the message "Unable to load network list",


• "Reset All Settings",


• Trying many mixes of restarting with cables in and out and every different USB port available to me,


• Offering my lunch to the gods.

The software involved is:

• iTunes 12.1.50


• iOS 8.1.2


• OS X 10.10.2

I've also checked and the phone is definitely unlocked, and marked as last activated yesterday when I first tried to use it.

I would be greatly appreciative of any further suggestions.