you'd be betting for value since he could call you with worse
but he doesn't have that much worse other than T9 type of hands

if you think he'd check a bad jack down to the river, then you kind of can't bet unless you think he'd lead it on the river
the board is kind of terrible since a backdoor flush got there, and you can't rule out that he might think you would check a flush draw down to bet it when it comes
of course he can have hands like 79 and other crap like that that just got a straight

but I don't know if we necessarily have enough information, some people call in this spot with a five or 77
so I'm most concerned with his calling range
if his calling range (of things we beat) is like 57s, 67s, 77, 99, 22-44 that's 3+3+6+6+18 = 36 combos
his calling range of things we lose to is JT, QJ, KJ, AJ = 9 + 12 + 12 + 9 = 42 combos
his raising range is all kinds of crap, but it doesn't matter because his calling range that beats us is already bigger (unless he leads hands like AJ on the river which we will have to calculate differently then)

but as you can see, there's really nothing that he can have other than random ace high calldowns that would make it EV+
even then, he's going to have hands that are worth a raise a lot and he might c/r a straight on the river because he hopes that you'll bluff after it got checked through 3 times