ETA (% E^-(% PI/SQRT (3))) = 27^(1/8)*GAMMA (1/3)^(3/2)/(2*2^(1/4)*(SQRT (3) + 2)^(1/8)*% PI)
eta((%e)^( - ((2 * %pi)/(sqrt(5))))) = ((sqrt(Gamma((1/20))) * sqrt(Gamma((9/20))))/(2^(7/8) * 5^(1/16) * (sqrt(5) + 1)^(3/8) * (%pi)^(3/4)))
ETA(%E^-(2*SQRT(2)*%PI/SQRT(3))) = GAMMA(1/24)*SQRT(SIN(%PI/24))*CSC(%PI/8)^(1/6)/(2^(23/24)*3^(1/8)*(SQRT(3)-1)^(1/4)*SQRT(%PI)*SQRT(GAMMA(1/12)))
ETA(%E^-(SQRT(3)*%PI)) = 3^(1/8)*(SQRT(3)+1)^(1/4)*GAMMA(1/3)^(3/2)/(2*2^(3/8)*%PI)
ETA(%E^-(SQRT(5)*%PI)) = 2^(1/8)*(2/%PHI^(3/2)+1)^(1/8)*SQRT(GAMMA(1/20))*SQRT(GAMMA(9/20))/(2*5^(1/16)*10^(1/4)*%PHI^(1/8)*%PI^(3/4))
ETA(%E^-(4*%PI/SQRT(3))) = 3^(3/8)*(SQRT(3)+1)^(1/4)*GAMMA(1/3)^(3/2)/(2*2^(1/8)*8^(1/4)*%PI)
eta((%e)^(-2*sqrt(5)*%pi))=((sqrt(Gamma((1/20)))*sqrt(Gamma((9/20))))/(2^(7/8)*5^(5/16)*(sqrt(5)+1)^(3/8)*(%pi)^(3/4)))
['AT('DIFF(THETA[1](Z,Q),Z,1),Z = 0) = 2*ETA(Q^2)^3,THETA[2](0,Q) = 2*ETA(Q^4)^2/ETA(Q^2),THETA[3](0,Q) = ETA(Q^2)^5/(ETA(Q)^2*ETA(Q^4)^2),THETA[4](0,Q) = ETA(Q)^2/ETA(Q^2)]
('diff(eta(q),q),%%=apply_nouns(%%),%%=subst([q=1/q,q=1/q],rhs(%%)))
'AT('DIFF(ETA(Q),Q,1),Q = %E^-(2*%PI/SQRT(7))) = -%E^(2*%PI/SQRT(7))*SQRT(GAMMA(1/7))*SQRT(GAMMA(2/7))*SQRT(GAMMA(4/7))*(9*7^(1/8)*GAMMA(1/7)^2*GAMMA(2/7)^2*GAMMA(4/7)^2/(32*SQRT(2)*%PI^3)-7*SQRT(2)/7^(3/8))/(32*%PI^2)
'at('diff(eta(q),q,1),q = %e^-(2*%pi/sqrt(3))) = 3*%e^(2*%pi/sqrt(3))*Gamma(1/3)^(3/2)*(32*%pi^3-2^(1/3)*sqrt(3)*Gamma(1/3)^6)/(512*2^(1/3)*3^(1/8)*%pi^5)
'at('diff(eta(q),q,1),q = %e^-(6*%pi/sqrt(3))) = %e^(6*%pi/sqrt(3))*Gamma(1/3)^(3/2)*(2^(1/3)*sqrt(3)*Gamma(1/3)^6+32*%pi^3)/(512*2^(1/3)*3^(3/8)*%pi^5)
'AT('DIFF(ETA(Q),Q,1),Q = %E^-(2*SQRT(7)*%PI)) = -%E^(2*SQRT(7)*%PI)*SQRT(GAMMA(1/7))*SQRT(GAMMA(2/7))*SQRT(GAMMA(4/7))*(-9*GAMMA(1/7)^2*GAMMA(2/7)^2*GAMMA(4/7)^2/(448*%PI^3)-1/SQRT(7))/(16*SQRT(2)*7^(1/8)*%PI^2)
'AT('DIFF(ETA(Q),Q,1),Q = %E^-(6*%PI)) = 2^(5/12)*%E^(6*%PI)*GAMMA(1/4)*(3^(3/4)*GAMMA(1/4)^4+18*SQRT(2)*(SQRT(3)-1)*%PI^2)/(1728*3^(3/8)*(SQRT(3)-1)^(5/6)*%PI^(15/4))
(('SUM(N/(1/Q^N-1),N,1,INF) = 'SUM(Q^N/(1-Q^N)^2,N,1,INF)) = 1/24-Q*'DIFF(LOG(ETA(Q)),Q,1)) = -('SUM(N*(3*N-1)*(-1)^N*Q^(N*(3*N-1)/2),N,MINF,INF))/(2*'SUM((-1)^N*Q^(N*(3*N-1)/2),N,MINF,INF))
'SUM(N/(1/Q^N-1),N,1,INF) = THETADERIV[1](%PI/3,2,Q^(1/6))/(24*SQRT(3)*ETA(Q))+1/24
x*('sum(n/(%e^(2*%pi*n*x)-1),n,1,inf)-1/24)+('sum(n/(%e^(2*%pi*n/x)-1),n,1,inf)-1/24)/x = -1/(4*%pi)
'Sum(n/(%e^(2*%pi*n/sqrt(7))-1),n,1,inf) = 9*Gamma(1/7)^2*Gamma(2/7)^2*Gamma(4/7)^2/(512*%pi^4)-sqrt(7)/(8*%pi)+1/24
'sum(n/(%e^(2*%pi*n/sqrt(5))-1),n,1,inf) = %phi*Gamma(1/20)^2*Gamma(9/20)^2/(192*5^(3/4)*%pi^3)-sqrt(5)/(8*%pi)+1/24
'Sum(n/(%e^(%pi*n)-1),n,1,inf) = Gamma(1/4)^4/(64*%pi^3)-1/(4*%pi)+1/24
'sum(n/(%e^(2*%pi*n/sqrt(3))-1),n,1,inf) = 3*2^(1/3)*Gamma(1/3)^6/(256*%pi^4)-3/(8*sqrt(3)*%pi)+1/24
'SUM(N/(%E^(2*%PI*N)-1),N,1,INF) = 1/24-1/(8*%PI)
'SUM(N/(%E^(3*%PI*N)-1),N,1,INF) = 1/24-(2-SQRT(2)*3^(1/4))^(1/4)*((2-SQRT(2)*3^(1/4))^(3/4)*(5*SQRT(2)*3^(3/4)+6*SQRT(3)+9*SQRT(2)*3^(1/4)+6)*GAMMA(1/4)^4/(288*(SQRT(3)-1)^(5/12)*%PI^2)+(SQRT(3)-1)^(7/12)/(2-SQRT(2)*3^(1/4))^(1/4))/(12*(SQRT(3)-1)^(7/12)*%PI)
'sum(n/(%e^(2*sqrt(3)*%pi*n)-1),n,1,inf)
= -2^(1/3)*Gamma(1/3)^6/(256*%pi^4)-1/(8*sqrt(3)*%pi)+1/24
'Sum(n/(%e^(2*sqrt(7)*%pi*n)-1),n,1,inf) = -9*Gamma(1/7)^2*Gamma(2/7)^2*Gamma(4/7)^2/(3584*%pi^4)-1/(8*sqrt(7)*%pi)+1/24
'sum(n/(%e^(6*%pi*n)-1),n,1,inf) = -sqrt(2)*3^(3/4)*Gamma(1/4)^4/(864*(sqrt(3)-1)*%pi^3)-1/(24*%pi)+1/24
'SUM(N/(%E^(6*%PI*N)-1),N,1,INF) = -SQRT(2)*3^(3/4)*GAMMA(1/4)^4/(864*(SQRT(3)-1)*%PI^3)-1/(24*%PI)+1/24
(n,d) =1, -d < 2 n <= d, (instead of -12 d < n < 12, due to exp(2 p n/d)1/24 having period 1 vs 24), where
g(n,d) := if d=1 then n else d*(floor(n/d)+3)-(g(d,nummod(n,d))*d+1)/nummod(n,d)
matrixmap(lambda([x],if numberp(x) and x<0 then concat('?-,-x)else x),addcol(transpose(append(["",d],1..20)),transpose(cons(n,makelist(?|,k,0,20))),addrow(matrix(-3..13),makelist(?-\-,k,-3,13),genmatrix(lambda([d,n],if ?numgcd(n,d)=1 then g(n,d)else""),20,13,1,-3))))
((1 = AGM(ETA(Q^2)^10/(ETA(Q)^4*ETA(Q^4)^4),ETA(Q)^4/ETA(Q^2)^2)) = AGM((16*ETA(Q^4)^8+ETA(Q)^8)^(5/12)/ETA(Q)^(2/3),ETA(Q)^(10/3)/(16*ETA(Q^4)^8+ETA(Q)^8)^(1/12))/ETA(Q^4)^(2/3)) = AGM(SQRT(64*ETA(Q^2)^24+ETA(Q)^24)-8*ETA(Q^2)^12,ETA(Q)^12)/(ETA(Q)^2*ETA(Q^2)^2*SQRT(SQRT(64*ETA(Q^2)^24+ETA(Q)^24)-8*ETA(Q^2)^12))
