Your talking about shooting pains, that normally is a nerve thing. A proper physio would be able to explain more but essentially, you have your nerves coming from your spine and when you get to the base of your spine you get a nerve called the sciatic nerve (hope I've spelt that right...!) this nerve runs all the way down your legs and if it gets trapped or irritated, just as with any nerve, it can become really painful.
I'd wonder if after an hour (guessing here) your posture has changed and the way your spine is being used has changed and as a result your nerves may not be as comfortable so one thing you could do is run with someone and ask them to spot-check you and point out any posture issues.
You do mention also that it seems to be coming from your knee, it could be a knee issue. Maybe the nerve is getting caught in something around your knee? (I've no idea if this is actually possible by the way!) and this would be an issue around your form and how you are running.
Things like that can often be dealt with either with looking at how your hips move the rest of your legs which could mean things like squats and lunges- with a focus on building up the muscles to support the correct form (let alignment and so on) or the way your ankles support the rest of your leg when you land on them.
Things around your ankles can be helped with making sure your balance and core stability is working well, making sure you have the correct footware on and your trainers are up to date (every 500 miles you are supposed to renew them) and if your running style is OK (some people have really poor running styles and it means they can feel very tired after a not very long because their bodies are doing more work then they need to). Focus on posture, head and chest up, arms and hands aiding your running not jazz hands or flopping about useless! A good friend could spot you again and don't forget that as you run, all these good posture things can and do change with tiredness and boredom.
I think it may be hard for your Doctor to find a trapped nerve, I don't think they run machines that do this (if they do, please let me know as I know someone who would find this very useful!) they just hazard a guess. A good osteopath however may be able to manipulate your spine and help things in some cases but they generally don't come cheap!
Sports massage could be another option if you can afford this as sometimes if a nerve has gotten caught they can help move things slightly enough to help but you'd probably need a few of these spread over a few months to get the full effects.
