$${\psi_n(z) = (-1)^{n+1} n! \ \zeta(n+1,z) }$$
$${\psi_n(1) = (-1)^{n+1} n! \ \zeta(n+1) }$$
$${\psi_{2n-1}(1) = (-1)^{n+1} B_{2n} \frac{2^{2n-2}}{n} \pi^{2n} }$$
$${\psi_n'(z) = \psi_{n+1}(z) }$$
$${B_0 = 1 }$$
$${B_m = - \frac{1}{m+1} \sum_{n=0}^{m-1} \binom{m+1}{n} B_n }$$
$${B_1 = - \frac{1}{2} }$$
$${B_2 = \frac{1}{6} }$$
$${B_3 = 0 }$$
$${B_4 = -\frac{1}{30} }$$
$${B_5 = 0 }$$
$${B_6 = \frac{1}{42} }$$
$${B_{2n+1} = 0, \ n \in \mathbb{Z}^+ }$$
$${ \frac{x}{e^x - 1} = \sum_{k = 0}^{\infty} \frac{B_k}{k!} x^k}$$
$${ x = (e^x - 1) \sum_{k = 0}^{\infty} \frac{B_k}{k!} x^k}$$
\({B_{n}'(x) = n B_{n-1}(x) }\)
\({ \displaystyle B_{n}(x) = \int{n B_{n-1}(x) \ dx } }\)
\({ \displaystyle \int_{0}^{1}{ B_{n}(x) \ dx = 0} }\)
$${\zeta(s,z) = \sum_{n = 0}^{\infty} \frac{1}{(n+z)^s} }$$
$${\zeta(s,1) = \zeta(s) }$$
$${ \zeta(2n) = \left \lvert B_{2n} \right \rvert \frac{2^{2n-1}}{(2n)!} \pi^{2n}, \ n \in \mathbb{Z}^+ }$$
$${ \zeta(-n) = - \frac{B_{n+1}}{n+1} , \ n \in \mathbb{Z}^+ }$$
$${ \zeta(-n,a) = - \frac{B_{n+1}(a)}{n+1} , \ n \in \mathbb{Z}^+ }$$
$${\zeta(0) = - \frac{1}{2} }$$
$${\zeta(1) = \gamma }$$
$${\zeta(2) = \frac{\pi^2}{6} }$$
$${\zeta(3) = 1.202... \text{Apery's constant} }$$
$${\zeta(4) = \frac{\pi^4}{90} }$$
$${\zeta(6) = \frac{\pi^6}{945} }$$
$${\zeta(-1) = - \frac{1}{12} }$$
$${\zeta(-2) = 0 }$$
$${\zeta(-3) = \frac{1}{120} }$$
$${\zeta(-4) = 0 }$$
$${\zeta(-5) = - \frac{1}{252} }$$
$${ \zeta(2, 2) = \sum_{k=0}^{\infty} \frac{1}{(k + 2)^2} }$$
$${ \zeta(2, 2) = \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \dots }$$
$${ \zeta(2, 2) = \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \dots }$$
$${ \zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \dots }$$
$${ \zeta(2, 2) = \zeta(2) - \frac{1}{1^2} }$$
$${ \zeta(2, 2) = \zeta(2) - 1 }$$
$${ \zeta(2, 2) = \frac{\pi^2}{6} - 1 }$$
$${\psi_2(1) = -2 \zeta(3) }$$
$${\psi_4(1) = -24 \zeta(5) }$$