I downloaded the minidump file and almost had a dump myself...no idea of how to read them. Well, for general troubleshooting I do general maintenance while looking for clues. For example is the computer and the fans clean inside or full of dust? Overheating parts can cause random shut downs. Have you installed anything or changed anything recently? A look in the 'Device Manager' should indicate if there are any easy major problems. There will be an exclamation mark on the part that is not working well. Is everything seated and connected well?
The hard drive has been checked o.k. so at most I would do a full chkdsk to ensure it is not writing to any bad memory giving a corrupt read and a bsod. then download memtest and check the DDR memory. An error there will cause a random bsod as well.
I have read the posts on L1c62x64.sys problems as well. If you know your machine has the "Atheros" network component, I would go into device manager, find the Atheros (I assume under Network Adapters) go for uninstall and reboot the computer. When it restarts it should take a second and re-find the component and install the drivers (assuming it all looked good in Device Manager when you looked first-no problems).
the last thing I can think of is the drivers. Do you have the motherboard chipset drivers installed, and update your other drivers. It seems excessive, but I reboot between updates even if not required as it reloads the registry with the new info. That's about it off the top of my head with no new info. Advice...your mileage may vary...I have an Intel Chipset and LAN so I find this useful:
http://www.intel.com/p/en_US/support/detect
Of course it can only tell you the Intel drivers that there are updates for.
If it still happens, need more details and perhaps someone smarter to look at this thread, like someone that can read the minidump