Closed form of Sum[Zeta[2k]/k^2,{k,1,Infinity}]

Questions
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1

How we can to obtain closed form to the

Sum[Zeta[2k]/k^2,{k,1,Infinity}]
2

Not yet at this time maybe in near future with AI .

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3

@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(2*n)-1)*4^n/n^2)

each programme give me another result

4

Using AI like ChatGPT or Gemini 3.1 pro :

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5

Good result is
3.319985937717828834823673110941457321592229779693541190919468911355429231390014421609241793964178613004986240132353652315479800855203283488633801842126438316056321894002418672404004800456936829260558

6

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

7

mildly related. used gemini 3.6f

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8
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