Using AI like ChatGPT or Gemini 3.1 pro :

Last equivalent

NIntegrate[(2 - 4 Pi x y Cot[2 Pi x y])/(x y) - (16 x y)/(
  1 - 4 x^2 y^2), {x, 0, 1}, {y, 0, 1}, WorkingPrecision -> 110] - 3.31998593771782883493971510040789760078738385062931491849491062371362\
85647769158186501992384950704057161955554 =
1.16041989466440279195154070935773727575441712358199333386901397040957\
4445308917927112093153*10^-19

only 18 digits is good

@Mariusz Iwaniuk could You find alternative sums or integrals for similar sum Sum[(Zeta[2 n] - 1) (4^n/n^2), {n, 1, Infinity}] Mathematica, Wolfram Cloud, Wolfram Alpha and pari procedure sumpos(n=1,(zeta(2n)-1)4^n/n^2)

each programme give me another result