The Beta function and the functional equation and pseudo inverse of matrix

$$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$
$$\xi(s) = \xi(1 - s)$$
$$\pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)=\pi^{-(1-s)/2}\ \Gamma\left(\frac{1-s}{2}\right)\ \zeta(1-s)$$
$$\frac{\pi^{-s/2}\ \pi^{-c/2}}{\pi^{-s/2-c/2+1/2}}\ \frac{\Gamma\left(\frac{s}{2}\right)\Gamma\left(\frac{c}{2}\right)}{\Gamma\left(\frac{s}{2}+\frac{c}{2}-\frac{1}{2}\right)}\ \frac{\zeta(s)\ \zeta(c)}{\zeta(s+c-1)}=\frac{\pi^{-(1-s)/2}\ \pi^{-(1-c)/2}}{\pi^{-((1-s)+(1-c)-(1-1))/2}}\ \frac{\Gamma\left(\frac{1-s}{2}\right)\ \Gamma\left(\frac{1-c}{2}\right)}{\Gamma\left(\frac{1-s}{2}+\frac{1-c}{2}-\frac{1-1}{2}\right)}\ \frac{\zeta(1-s)\ \zeta(1-c)}{\zeta((1-s)+(1-c)-(1-1))}$$
$$\frac{\zeta(s)\ \zeta(c)}{\zeta(s+c-1)}$$


Pseudo inverse of von Mangoldt function matrix or Ceta(s,c) as it could be called.

(*start*)
(*fungerande 8.10.2016 klockan 1:31 Riemann zeta zero spectrum*)
Clear[n, k, nn, s, c, M, A, B, a, t]
a = 4;
nn = 42;
Print["set s=0 and c=1+epsilon, or, s=1+epsilon and c=0 to get a \
matrix that corresponds to zeta zero spectrum"]
s = 8;
Print["s=", s]
c = 9;(*c=1+1/100000... ger matrisen för Riemann zeta zero spectrumet*)
\

Print["c=", c]
M = Inverse[
   Table[Table[If[Mod[n, k] == 0, 1/(n/k)^s, 0], {k, 1, nn}], {n, 1,
     nn}]];
A = Inverse[
   Table[Table[If[Mod[n, k] == 0, k^c, 0], {k, 1, nn}], {n, 1, nn}]];
B = Transpose[
   Table[Table[If[Mod[n, k] == 0, k/n^s, 0], {k, 1, nn}], {n, 1, nn}]];
Print["the matrix M.M.A.B where A.B is the Ramanujan sum matrix and M \
is the Möbius function matrix"]
Print["M.M.A.B is the pseudo inverse of the von Mangoldt function \
matrix"]
MatrixForm[(M.M.A.B)[[1 ;; 12, 1 ;; 12]]]
Print["sum of the ", a, "-th column"]
(Transpose[Transpose[M.M.A.B*Range[nn]^(c*0)]*Range[nn]^(s*0)])[[All,
   a]];
N[Total[%], 30]
Print["generating function of the ", a, "-th column"]
N[(Total[Divisors[a]^(1 - c)])/Zeta[c]/Zeta[s]^2/a^s, 30]
Print["sum of the whole matrix"]
Total[Total[N[M.M.A.B, 30]]]
Print["generating function for the whole matrix"]
N[Zeta[s + c - 1]/(Zeta[s]*Zeta[c]), 30]
Print["these differences gives a hint at what the row generating \
functions should be"]
(((Transpose[Transpose[A.B*Range[nn]^(c*1)]*Range[nn]^(s*1)])[[a,
     All]]) - ((Transpose[
      Transpose[M.M.A.B*Range[nn]^(c*1)]*Range[nn]^(s*1)])[[a, All]]))
(*end*)