Here is a better approximation to the so far known value of Viswanath's constant from the MRB constant and Pi. When more digits of Viswanath are discovered, the approximations will have to be changed!

v = 1.1319882487943
(*Let v be a known approximation to Viswanath's constant.*)
m = NSum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 200];
(*Let m be a known approximation to the MRB constant.*)

Rationalize[v, 0] - (53319 Pi- 98596 m)/131613 
(*-4.2569622784754151216498253699...2*10^-17*)`

(53319 Pi - 98596 m)/131613==1.13198824879430007...