There could be several reasons for this.
No fuses (at last not in my SL3’s) so it ain’t that.
First, no power, but you say you’ve checked that (disconnected and reconnected the power cable, right?).
From here on out, it requires opening the back of the speaker. But you’re probably going to need to do this anyway, because even if you send something off to service, it’s better to send the crossover board and not the whole unit.
There are several parts that could be at fault:
First, is the wiring from the EIC power socket to the main board, it terminates into a screw strip. Make sure that’s well screwed in and making solid contact.
Next is the music sense, a circuit that detects voltage on the speaker input terminals and ‘turns on’ the AC power to the rest of the unit.
If the wires from the binding posts are not making good contacts, then this might not trigger.
This is provided via a gray wire from the binding posts to the circuit board, check both ends for good connections.
Beyond those two simple things, it could be any number of more complex things, and I’d also refer you to service for those.
But several times, we’ve seen loose wires be the culprits for non-functioning. So checking for that is simple and quick. Well, quicker than shipping something off
Search and read up on various SL3 modification threads for pics of the boards and the open units.