Check your mail servers cert using a tls connection:

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- Careful Chrome users, this search box might be "Not secure"

See this post by koppor on
If that site or link goes away, here are the gory details:
apt-get install gnutls-bin to have gnutls-cli available.
gnutls-cli -s -p 25 YOURSMTPHOST (starts a session with your mailserver)
ehlo foo
Press CTRL+d (^d)

This one works as well:
Thanks to Dan Andreatta, and Skyhawk (for the edits):
Again, in case that goes away:
openssl s_client -connect -starttls smtp
or standard secure smtp
openssl s_client -connect