outside deck stairs

I've installed stairs for my outside deck which the treads are 48" wide and supported by 2x12 stringers
on either side attached to Simpson stair angles [10"]. These stair treads are made of 4x12 lumber.
Since these are 4x's do I still need a central support stringer mid-way between the 48" span?

If the stairs feel solid, they are fine. 4X12s can handle a lot of weight. Just make sure the step height doesn't exceed the recommended rise/run ratios. Normally an 8" step height is about right, but mostly all the steps need to be the same height.