Help7LM:=proc(): print(`JGlm1(N), ..., JGlm5(N) `): end:
#Start From Lucy
##The following procedure agrees to 576 digits if you stop after n=100
JGlm1:=proc(N) local a,n:
a:=add((-1)^n*(6*n)!/(n!)^6*(5418*n^2+693*n+29)/(2880^(3*n)),n=0..N):
sqrt((128*sqrt(5))/a):
end:

#The following sum comes from page 4 of https://arxiv.org/pdf/2503.00570


##The following procedure agrees to 365 digits if you stop after n=100
JGlm2:=proc(N) local a,n:
a:=add((43680*n^4+20632*n^3+4340*n^2+466*n+21)*(1/2^(12*n))*(1/(n!)^9)*RF(1/2,n)^7*RF(1/4,n)*RF(3/4,n),n=0..N):
(2048/a)^(1/4):
end:

#The following 3 sums come from the paper:
# https://arxiv.org/pdf/1104.0392


##The following procedure agrees to 170 digits if you stop after n=100
JGlm3:=proc(N) local a,n:
a:=add((-1)^n*binomial(4*n,2*n)*binomial(2*n,n)^2*(28*n+3)/(3^n*2^(12*n)),n=0..N):
16*sqrt(3)/(3*a):
end:


##The following procedure agrees to 96 digits if you stop after n=100
JGlm4:=proc(N) local a,n:
a:=add(binomial(4*n,2*n)*binomial(2*n,n)^2*(8*n+1)/(2^(8*n)*3^(2*n)),n=0..N):
2*sqrt(3)/a:
end:


##The following procedure agrees to 61 digits if you stop after n=100
JGlm5:=proc(N) local a,n:
a:=add(binomial(2*n,n)^3*(6*n+1)/2^(8*n),n=0..N):
4/a:
end:

#End from Lucy