http://algo.inria.fr/csolve/mrtns.pdf  pg4  (Steven Finch)


with 	K = LandauRamanujanK
	G = Catalan

ne=Product[p=Prime[q];w=If[Mod[p,4]===3,p,Infinity];1-1/w^2,{q,1,Infinity}]==1/2/K^2
   (1-1/3^2)*(1-1/7^2)*(1-1/11^2)*(1-1/19^2)*(1-1/23^2)*(1-1/31^2)*..

no=Product[p=Prime[q];w=If[Mod[p,4]===1,p,Infinity];1-1/w^2,{q,1,Infinity}]==16 K^2/Pi^2
   (1-1/5^2)*(1-1/13^2)*(1-1/17^2)*(1-1/29^2)*(1-1/37^2)*(1-1/41^2)*..

pe=Product[p=Prime[q];w=If[Mod[p,4]===3,p,Infinity];1+1/w^2,{q,1,Infinity}]==Pi^2/16/G/K^2
   (1+1/3^2)*(1+1/7^2)*(1+1/11^2)*(1+1/19^2)*(1+1/23^2)*(1+1/31^2)*..

po=Product[p=Prime[q];w=If[Mod[p,4]===1,p,Infinity];1+1/w^2,{q,1,Infinity}]==192 G K^2/Pi^4
   (1+1/5^2)*(1+1/13^2)*(1+1/17^2)*(1+1/29^2)*(1+1/37^2)*(1+1/41^2)*..

given that (* http://library.wolfram.com/infocenter/Demos/120/ *)

LandauRamanujan[n_] := With[{K = Ceiling[Log[2, n*Log[3, 10]]]}, 
   N[1/Sqrt[2]*Product[(((1 - 2^(-2^k))*4^2^k*Zeta[2^k])/
        (Zeta[2^k, 1/4] - Zeta[2^k, 3/4]))^2^(-k - 1), {k, 1, K}], n]]

ne -> 0.85610898172189347690603
no -> 0.94680640718007933421609
pe -> 1.15308056158544787036526
po -> 1.05443994479994848964882

or to any (reasonably) desired accuracy and precision.