Results of our model with a near-solar threshold at Z_th = 8.6 (i.e. at ~0.6 Z_⊙), where κ = 20 (i.e. simulating a sharp metallicity cutoff) and the results are normalised to the LGRB rate estimate of Ghirlanda & Salvaterra (2022) so that ${\eta }_0=4.91\cdot {10}^{-5}{M}_{\odot }^{-1}$. Panels A and B show the Z_O/H,i vs z_i (with bins of Δz = 0.2 and ΔZ_O/H = 0.1) and M_*,i vs z_i distributions (with bins of Δz = 0.2 and Δ log M_* = 0.1 dex), respectively, corresponding to dM/dt_o (Eq. (17)) and divided by the bin sizes Δz = 0.2 and Δ log y = Δ log Z = Δ log M_* = 0.1 to make the values independent of bin size choice. In these distributions we set all values less than 0.01% of the maximum density to zero. Panel B also contains M_lim (dashed line, Eq. (1)). Panel C shows the integrated distributions shown in panels A and B (i.e. integrated over Z_O/H and M_*, respectively), but without the dt_s/dt_o · dV factors in Eq. (17) in order to make a comparison with the CSFH of Madau & Dickinson (2014), corrected to a Kroupa (2001) IMF (following Appendix B of Chruślińska & Nelemans 2019). Panels D and E show distributions similar to panels A and B, but instead weighted by R_LGRB/Δz/Δ log y. For a comparison with observation, we show the C17 metallicity estimates of Graham et al. (2023) and the mass estimates of Perley et al. (2016c) in the respective panels, where the former is valid for z ≤ 2.5 (shown with a dash-dotted grey line) and the latter contains constrained estimates (circles) as well as upper limits (triangles). Panel F, in turn, shows the integrated R_LGRB distribution, similarly to panel C without the dt_s/dt_o · dV factors, effectively resulting in a LGRB density (ρ_LGRB), together with the results of Ghirlanda & Salvaterra (2022, dark grey line) and its 1σ uncertainty (shaded region). Panel G shows a normalised histogram (with bins of 0.15) corresponding to the Graham et al. (2023) metallicities, which are valid for z ≤ 2.5, and the distributions from panels A and D, integrated over redshift within this redshift region and normalised (red and blue lines, respectively). In panels D and G we also show Z_th (dotted line). Finally, panel H shows a normalised histogram (with bins of 0.5 dex) of the host stellar masses in our local LGRBs sample (Appendix A, different from the data shown in panel E), valid for z ≤ 0.5, together with the corresponding distributions from panels B and E that were integrated over redshift for z ≤ 0.5 and normalised (red and blue lines, respectively). The dotted grey line in the top row shows solar metallicity.