Check out these photos -- obtained by TMZ -- that show the First Son at Dan Tana's, an Italian restaurant in West Hollywood, Thursday night and, he looked pretty chill for a guy facing a long prison sentence ... waiting patiently in the lobby along with all the normies.

Hunter dressed casually for the occasion -- ripped jeans and a polo ... as opposed to the full suit we've grown accustomed to seeing him during his trial in Delaware.

While he's hanging out alone in these pics here, we're told HB actually went to Dan Tana's with his wife -- Melissa -- last night ... as well as a member of his security team.

Eyewitnesses tell us Hunter appeared in good spirits, chatting with his security about the NBA Finals before his bodyguard posted up at the front door for the night. Melissa -- on the other hand -- appeared a bit more serious, our sources say.

She has a good reason ... her husband's facing serious time behind after a jury convicted him on 3 felony counts of lying on a federal firearms application -- up to 25 years, in fact.

While that's the maximum sentence on the table, Hunter will likely receive a much more lenient sentence. President Joe Biden said he's proud his son's clean now and will respect the outcome of the case.