Beta and rho overlap from roughly 405 nm - 540 nm, and for most of that range, gamma will also produce a signal. The only place that we can get beta and rho without gamma is 405 nm - 425 nm. And in that range, beta is much more sensitive than rho.
There is no single wavelength in the visible spectrum that will stimulate equal signals from both beta and rho, without gamma.
But what if you mixed two beams of light - one around 405 nm (which would stimulate only beta), and one from 675 nm (which would stimulate only rho)? With this setup, you can jam into the brain combinations of signals that do not occur with pure spectral colors. And in this case, the brain perceives the combination as pink (if it's a light shade) or purple (if dark).
In addition to generating colors that are not found in the spectrum, you can fake colors that do exist in the spectrum. Shine 625 nm light into the eye, the photoreceptors in the eye get "mostly rho, with one quarter as much gamma", which the brain perceives as the color orange. But the brain will perceive the same color orange with a mixed beam of yellowish-green 550 nm (which stimulates more gamma than rho), sweetened up with a shot of 675 nm red (pure rho). [When two spectra are different, but look alike to the observer, they are called "metamers" or "monomers".]
When all of the cones are stimulated, the brain provides the sensation of "white". You also see white when presented with an extremely bright light of any color. This may be because, with the overlap of color sensors, a very bright light source will produce strong signals in all receptors.
You might remember back to your primary school art class, when Miss Arglebargle said "yellow and red make orange". Well, they don't make real orange. A prism can show the difference between Arglebargle Orange and the real thing. But they make something that fools your brain into perceiving orange - and that's good enough.
This is also the origin of the three "primary colors" that can be mixed to produce other colors. The colors work that way because the eye has light receptors tuned to the three primary wavelengths, and the brain perceives mixtures of these three stimulus wavelengths as a single different color.