Integral and Series Representations of Riemann's Zeta Function and Dirichlet's Eta Function and a Medley of Related Results

Table 1

The first few values of 𝜅(0,𝜅+1,𝜋²/𝜔²) in (93).

𝑘	Analytic	𝜔=1	𝜔=𝜋
0	−12+𝜔coth(𝜔)2	0.156518	1.076674
1	−12+14𝜔^2sinh^2𝜔+14𝜔coth(𝜔)	0.927424 × 10^−2	0.306837
2	3𝜔16coth^31𝜔−2coth^2𝜔+𝜔coth𝜔−3+2𝜔^216sinh^2𝜔+8+3𝜔^216sinh^2𝜔	0.795491 × 10^−3	0.134296