There is no known solution for this problem, all I can suggest is the following:
- Check
https://accounts.legionsoverdrive.com/ in Internet Explorer and see if that loads without any warnings, if you get any certificate warnings - you've got problems.
- Reinstall .NET framework, as this is what the launcher uses.
- If you've got any anti-virus or firewall program installed then try disabling it, as it might not be happy with the fact that the launcher uses SSL.
- Make sure your Windows is fully patched, SP3 on XP, SP2 on Vista, SP1 on Win7 - and make sure it has every single security patch installed after that.
- Ensure your clock is accurate, the right date/time etc.