Heavy warships/quinquiremes/Juggernauts are ships that deal 30 crush + 30 hack damage per shot, which increases if you garrison catapults. The problem is, they are useless. I was recently in a game that proved how weak the heavy warship is - at huge expense I produced a heavy warship with 6 catapults in it, which was able to destroy only a couple of docks before the enemy chariots arrived and forced it to retreat. It was not able to kill any towers or fortresses, which was my intended purpose of making the catapults and warship.
Here are the stats for Seleucid trireme:
1400 HP, 35 pierce damage per 2 seconds per arrow, armor: 5 hack, 10 pierce, 5 crush, range 55
For a Seleucid heavy warship:
2000 HP, 30 hack + 30 crush per 5 seconds per shot, armor: 5/10/5, range 65
Without garrison, they are useless in ship-to-ship combat. If you garrison a trireme-class ship with 10 civilian soldiers - which you can do in age II, as soon as you make a trireme - it will destroy an empty heavy warship.
Producing a garrisoned heavy warship is an incredibly expensive late game luxury. For Seleucids, 1 heavy warship with 5 catapults in it costs 1950 wood 1750 stone 200 metal. It is more expensive than two fortresses - approaching the cost of three fortresses - and occupies 18 population. Plus, catapults require fortresses to produce them.
For ship-to-ship combat, a garrisoned heavy warship will not necessarily beat a garrisoned trireme, in addition to the crazy expense. The trireme with 10 citizen-soldiers deals 35/2*13 = 227 pierce damage per second, which is 80 damage per second after ship armor. The heavy warship with 5 catapults deals 30/5*6 = 36 hack plus 36 crush damage per second, which is 43 damage per second after ship armor. Even though the garrisoned heavy warship has more HP, it only deals about half as much damage.
Against buildings, pierce damage is negligible due to armor. That 36 hack + 36 crush damage per second, reduced by a fortress's 15 hack armor and 2 crush armor, comes out to 36.5 damage per second. At this rate it would take about 2 minutes to take down a 4200 HP fortress with the heavy warship. This rate is comparable to two single catapults not on a ship. It is very slow, giving the opponent plenty of time to react, or simply to repair the fortress. At 4.2 HP/second/worker, nine repairing workers would be sufficient to prevent the fortress from losing HP.
Now consider, as another form of naval siege, a Briton medium warship with 5 battering rams in it. They could unload on the shore and kill the same fortress in roughly 10 seconds (4200 damage / 500 damage per second), plus the time it takes the rams to maneuver into position, which doesn't take long if the fortress is on the shore. Cost: 1900 wood 1150 metal.
So what change could balance the heavy warship? Let's operate under the premise that it ought to be at least good for naval siege. For this, have 100 crush damage per shot instead of 30, and have 10 pierce damage instead of 30 hack damage. The catapults do 100 crush and 10 pierce damage while not in the ship if they aren't upgraded, so shouldn't they do the same damage while in the ship? At this rate, it would take about 43 seconds to destroy the fortress. That's still slower than unloading rams, but it's maybe in the right area. I calculate that if the heavy warship has both armor upgrades (so 16 pierce armor) and the fortress is shooting 23 arrows, it would kill the heavy warship in about 48 seconds, which is good because it means the garrisoned heavy warship would kill the fortress before the fortress kills the garrisoned heavy warship.