We present SLASH, a pitch estimation method of speech signals based on self-supervised learning (SSL). To enhance the performance of conventional SSL-based approaches that primarily depend on the relative pitch difference derived from pitch shifting, our method incorporates absolute pitch values by 1) introducing a prior pitch distribution derived from digital signal processing (DSP), and 2) optimizing absolute pitch through gradient descent with a loss between the target and differentiable DSP-derived spectrograms. To stabilize the optimization, a novel spectrogram generation method is used that skips complicated waveform generation. In addition, the aperiodic components in speech are accurately predicted through differentiable DSP, enhancing the method's applicability to speech signal processing. Experimental results showed that the proposed method outperformed both baseline DSP and SSL-based pitch estimation methods, attributed to the effective integration of SSL and DSP.