Hello, Albert

From the post "$\tau_o$ Or No $\tau_o$",

$\tau_o\ne\cfrac{c^4}{4\pi G}$

but we can have,

$\tau_o=-A\cfrac{c^4}{4\pi G}$

since  $\tau_o$  is already unit dimensionally consistent,  $A$  is a dimensionless constant.  And if we are really cheeky,

$\tau_o=-\Lambda \cfrac{c^4}{8\pi G}$

where   $A=\cfrac{\Lambda }{2}$.

Hello Albert Einstein!  Unfortunately,  Einstein's  $\Lambda$  has unit  m^-2, it is likely that he moved from radial formulation to volumetric enclosure and needed  $\Lambda$  to scale from m^-1 to m^-3.