Here's an example with a faster-firing weapon (so that there is less rounding error) at three different points as you vary the number of servos.

First, here is the turret with reduced servos until it no longer overheats.

The turret is a triple turret, so the number of barrels is 3. Using the non-overheating turret formula,

DPS = Nb * D * R

where Nb is the number of barrels (i.e., double, triple, quad), D is the damage, and R is the fire rate, the DPS should be:

Nb*D*R = 3*2640.4*1.5 = 11881.8

(Yes, I know many of you believe the number of barrels to be meaningless, but please stay with me on this...)

However, the displayed DPS is VERY close to 1/3 of that value, suggesting that the number of barrels is meaningless:

Dispalyed DPS = 3913.9

11882/3 (or simply 2640.4*1.5) = 3960.6

I believe that the difference between these two numbers is caused by rounding error on the firing rate. The actual firing rate of the weapon is probably 3913.9/2640.4 = 1.482, which of course would round up to the displayed value of 1.5.

NOW, I add just ONE more servo so that the weapon now can overheat, and the displayed DPS jumps by a factor of 3!:

Now that the turret overheats, I will use the overheating turret formula:

DPS = Nb * D * R * Ft / (Ft + Ct)

where Ft is the continuous firing time, and Ct is the cooldown time. So I calculate the turret should have a DPS of:

DPS = Nb * D * R * Ft / (Ft + Ct) = 3*2640.4*1.7*104.2/(104.2+5.6) = 12779.25

which matches the in-game displayed value 12534.7 pretty closely. Again, the difference is probably caused by rounding error on the displayed firing rate.

Finally, this is what the turret looks like with the maximum number of servos:

As you can see, now that the weapon is cooldown-limited, increasing the number of servos only increases the burst DPS; it doesn't affect the weapon's average damage over time (which is the displayed value). So calculating the DPS in this scenario, I get:

DPS = Nb * D * R * Ft / (Ft + Ct) = 3*2640.4*3.7*4.2/(4.2+5.6) = 12560.76

which is significantly closer the the in-game displayed value of 12534.7. Again, the difference is probably caused by rounding error.

So I ask again: What is the turret's actual DPS? Because it seems very unlikely to me that switching a turret from non-cooldown to cooldown should suddenly cause the DPS to increase by the a factor equal to the number of barrels the weapon has. (If it DOES, then that itself it likely a bug).

Based on my gameplay experience, I think that most likely, the number of barrels is meaningless and the displayed DPS value for any turret with a cooldown is incorrect by a factor equal to the number of barrels. Slightly less likely to me is that the number of barrels DOES matter and the displayed DPS values for turrets without a cooldown is wrong.