If something is moving past me at a relativistic speed, does its gravitational force on me increase as its speed increases? That is, not as if its speed is changing, but if multiple of the same (rest) mass go past me in the same path, is the gravitational attraction at the closest point the same for each, or is it higher the faster it's moving?

As some background, I was recently surprised to see that I should discard the idea of "relativistic mass", so I've been trying to work on that. Please forgive me mixing Newtonian physics with relativity, but I'm trying to get a high level understanding before going into the math too much. I understand now that instead of $F=m\_{relativistic}a$ I should use $F=\frac{d}{dt}(\gamma m\_0v)$, but what about $F=\frac{Gm\_1m\_2}{r^2}$? Is that still roughly as it is, or do I need to throw a factor of $\gamma$ in there, or something else entirely?

Yes, I know this is talking about acceleration and so it doesn't really fit into special relativity, and lots of other disclaimers, but I'm hoping for a simple answer to get a general understanding.