How I did verify my bitcoin 0.13 download.
Since there is this scary warming at the top of the page at bitcoin.org I had to check my downloads from the site.
I got both bitcoin-0.13.0-win64-setup.exe and bitcoin-0.13.0-win64.zip and run the sha256sum on them.
I didn't know the right results. I just searched with the resulting hash the web and looked at the links they produce.
They were just fine.
For the history these are the results I got:
That's exactly how you do it.