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Understanding the physical processes sculpting the appearance of young gas-giant planets is complicated by degeneracies confounding effective temperature, surface gravity, cloudiness, and chemistry. To enable more detailed studies, spectroscopic observations covering a wide range of wavelengths is required. Here we present the first L-band spectroscopic observations of HR 8799 d and e and the first low-resolution wide bandwidth L-band spectroscopic measurements of HR 8799 c. These measurements were facilitated by an upgraded LMIRCam/ALES instrument at the LBT, together with a new apodizing phase plate coronagraph. Our data are generally consistent with previous photometric observations covering similar wavelengths, yet there exists some tension with narrowband photometry for HR 8799 c. With the addition of our spectra, each of the three innermost observed planets in the HR 8799 system have had their spectral energy distributions measured with integral field spectroscopy covering $\sim0.9$ to $4.1~μ\mathrm{m}$. We combine these spectra with measurements from the literature and fit synthetic model atmospheres. We demonstrate that the bolometric luminosity of the planets is not sensitive to the choice of model atmosphere used to interpolate between measurements and extrapolate beyond them. Combining luminosity with age and mass constraints, we show that the predictions of evolutionary models are narrowly peaked for effective temperature, surface gravity, and planetary radius. By holding these parameters at their predicted values, we show that more flexible cloud models can provide good fits to the data while being consistent with the expectations of evolutionary models.