ABSTRACT We study the statistics of weak lensing convergence peaks, such as their abundance and two-point correlation function (2PCF), for a wide range of cosmological parameters Ωm and σ8 within the standard ΛCDM paradigm, focusing on intermediate-height peaks with signal-to-noise ratio (SNR) of 1.5–3.5. We find that the cosmology dependence of the peak abundance can be described by a one-parameter fitting formula that is accurate to within $∼ 3{{\ \rm per\ cent}}$. The peak 2PCFs are shown to feature a self-similar behaviour: if the peak separation is rescaled by the mean interpeak distance, catalogues with different minimum peak SNR values have identical clustering, which suggests that the peak abundance and clustering are closely interconnected. A simple fitting model for the rescaled 2PCF is given, which together with the peak abundance model above can predict peak 2PCFs with an accuracy better than $∼ 5{{\ \rm per\ cent}}$. The abundance and 2PCFs for intermediate peaks have very different dependencies on Ωm and σ8, implying that their combination can be used to break the degeneracy between these two parameters.