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Statements
ζ
′
(
s
,
a
)
=
-
ln
a
a
s
(
1
2
+
a
s
-
1
)
-
a
1
-
s
(
s
-
1
)
2
+
s
(
s
+
1
)
2
∫
0
∞
(
B
~
2
(
x
)
-
B
2
)
ln
(
x
+
a
)
(
x
+
a
)
s
+
2
d
x
-
(
2
s
+
1
)
2
∫
0
∞
B
~
2
(
x
)
-
B
2
(
x
+
a
)
s
+
2
d
x
,
diffop
Hurwitz-zeta
1
𝑠
𝑎
𝑎
superscript
𝑎
𝑠
1
2
𝑎
𝑠
1
superscript
𝑎
1
𝑠
superscript
𝑠
1
2
𝑠
𝑠
1
2
superscript
subscript
0
periodic-Bernoulli-polynomial-B
2
𝑥
Bernoulli-number-B
2
𝑥
𝑎
superscript
𝑥
𝑎
𝑠
2
𝑥
2
𝑠
1
2
superscript
subscript
0
periodic-Bernoulli-polynomial-B
2
𝑥
Bernoulli-number-B
2
superscript
𝑥
𝑎
𝑠
2
𝑥
{\displaystyle{\displaystyle\zeta'\left(s,a\right)=-\frac{\ln a}{a^{s}}\left(%
\frac{1}{2}+\frac{a}{s-1}\right)-\frac{a^{1-s}}{(s-1)^{2}}+\frac{s(s+1)}{2}%
\int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)-B_{2})\ln\left(x+a%
\right)}{(x+a)^{s+2}}\mathrm{d}x-\frac{(2s+1)}{2}\int_{0}^{\infty}\frac{%
\widetilde{B}_{2}\left(x\right)-B_{2}}{(x+a)^{s+2}}\mathrm{d}x,}}
ℜ
s
>
-
1
𝑠
1
{\displaystyle{\displaystyle\Re s>-1}}
s
≠
1
𝑠
1
{\displaystyle{\displaystyle s\neq 1}}
a
>
0
𝑎
0
{\displaystyle{\displaystyle a>0}}
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)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
ln
z
𝑧
{\displaystyle{\displaystyle\ln\NVar{z}}}
B
~
n
(
x
)
periodic-Bernoulli-polynomial-B
𝑛
𝑥
{\displaystyle{\displaystyle\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
x
𝑥
{\displaystyle{\displaystyle x}}
a
𝑎
{\displaystyle{\displaystyle a}}
s
𝑠
{\displaystyle{\displaystyle s}}
k
𝑘
{\displaystyle{\displaystyle k}}
Identifiers
