Converting "Luminosity per unit wavelength" to "Luminosity per unit frequency"

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If there is a luminosity per unit wavelength defined as L_λ dλ, am I correct in thinking that the conversion to luminosity per unit frequency (L_ν) would be L_ν = L_λ (λ^2/c)?

And building off of that, if the shape of a spectrum was proportional to a power-law: L_λ = λ^(β), is there a way to find an analytical expression for L_ν in terms of β? From plugging the above in, would it be accurate to say that L_ν = ν^(β-2)?

Note: this isn't for homework or anything--I'm just looking for an easy way to compare how spectral shape would change based on the way the spectrum is defined.

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That would be Lν ~ ν-(β+2)
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