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Originally Posted by Calvin
Don't exist at all? Or do they have the same instability that the L1 through L3 normally have? The plan is for two 1.2 solar mass stars and one 0.6 solar mass star. I don't suppose there's some other set of gravitationally stable points to use?
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Don't exist at all. An object at L4 or L5 is orbiting the primary (larger) mass, with just enough perturbation from the secondary to keep it fixed in place (in a co-rotating coordinate system). If the secondary is too large, the object essentially "doesn't know" which mass to orbit and wanders around instead.
Your system will have long-term stable restricted four-body (three stars + object) orbits, but they won't be confined to particular points in space. In fact, they probably wander around all three stars, taking tens or hundreds of thousands of years to make the circuit.
The Lagrange points are very much a special case in astrodynamics, although one that turns up fairly frequently in real life.