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Statements
ζ
′
(
1
-
2
n
,
1
3
)
=
-
π
(
9
n
-
1
)
B
2
n
8
n
3
(
3
2
n
-
1
-
1
)
-
B
2
n
ln
3
4
n
⋅
3
2
n
-
1
-
(
-
1
)
n
ψ
(
2
n
-
1
)
(
1
3
)
2
3
(
6
π
)
2
n
-
1
-
(
3
2
n
-
1
-
1
)
ζ
′
(
1
-
2
n
)
2
⋅
3
2
n
-
1
,
diffop
Hurwitz-zeta
1
1
2
𝑛
1
3
𝜋
superscript
9
𝑛
1
Bernoulli-number-B
2
𝑛
8
𝑛
3
superscript
3
2
𝑛
1
1
Bernoulli-number-B
2
𝑛
3
⋅
4
𝑛
superscript
3
2
𝑛
1
superscript
1
𝑛
digamma
2
𝑛
1
1
3
2
3
superscript
6
𝜋
2
𝑛
1
superscript
3
2
𝑛
1
1
diffop
Riemann-zeta
1
1
2
𝑛
⋅
2
superscript
3
2
𝑛
1
{\displaystyle{\displaystyle\zeta'\left(1-2n,\tfrac{1}{3}\right)=-\frac{\pi(9^%
{n}-1)B_{2n}}{8n\sqrt{3}(3^{2n-1}-1)}-\frac{B_{2n}\ln 3}{4n\cdot 3^{2n-1}}-%
\frac{(-1)^{n}{\psi^{(2n-1)}}\left(\frac{1}{3}\right)}{2\sqrt{3}(6\pi)^{2n-1}}%
-\frac{\left(3^{2n-1}-1\right)\zeta'\left(1-2n\right)}{2\cdot 3^{2n-1}},}}
n
=
1
,
2
,
3
,
…
𝑛
1
2
3
…
{\displaystyle{\displaystyle n=1,2,3,\dots}}
B
n
Bernoulli-number-B
𝑛
{\displaystyle{\displaystyle B_{\NVar{n}}}}
ζ
(
s
,
a
)
Hurwitz-zeta
𝑠
𝑎
{\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
ζ
(
s
)
Riemann-zeta
𝑠
{\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
π
{\displaystyle{\displaystyle\pi}}
ψ
(
z
)
digamma
𝑧
{\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
ln
z
𝑧
{\displaystyle{\displaystyle\ln\NVar{z}}}
n
𝑛
{\displaystyle{\displaystyle n}}
Identifiers
