This article has been edited for a bounty.
$C$ MRB, the MRB constant, is defined at http://mathworld.wolfram.com/MRBConstant.html .
There is an excellent 56 page paper whose author has passed away. You can find it in Google Scholar "MRB constant," Better yet, use the following link http://web.archive.org/web/20130430193005/http://www.perfscipress.com/papers/UniversalTOC25.pdf. You find a cached copy there.
Just before the author, Richerd Crandall, died I wrote him about a possible small error. What I'm worried about is formula 44 on page 29 and below. When I naively worked formula 44 it needed a negative sign in front of it. Crandall did write me back admitting to a typo, but he died before he had a chance to correct it. Is there anyone out there competent enough to check, correct and prove the corrected formulas for me? Thank you. I will use the proofs often and try to get the formulas published more.
Here is how I worked formula 44 and got -B:
(*define the Dirichlet eta function*)
eta[s_] := (1 - 2^(1 - s)) Zeta[s];
(*define the higher derivatives of the eta(0)*)
a[i_] := Derivative[i][eta][0];
(*Define c:*)
c[j_] := Sum[Binomial[j, d](-1)^dd^(j - d), {d, 1, j}]
(*formula (44)*)
N[Sum[c[m]/m!*a[m], {m, 1, 40}], 100]