Show[Plot[Zeta[s],{s,-10,10},PlotRange->{{-10,10},{-1,2}}]
,
Graphics[Point[{2,Pi^2/6}]]
,
Graphics[Point[{-1,-1/12}]]
]

Plot[Zeta[s],{s,-10,10},PlotRange->{{-10,1},{-.1,.1}}]

Zeta'[0]/Zeta[0]

Plot[{Sum[MangoldtLambda[n],{n,1,x}]
,
x-Log[2Pi]
}
,{x,1.2,20}]

ZetaZero[1]//N
ZetaZero[2]//N
ZetaZero[3]//N

kthContr[k_,x_]:=x^ZetaZero[k]/ZetaZero[k]+
x^(1-ZetaZero[k])/(1-ZetaZero[k])

Plot[kthContr[1,x],{x,1,20}]

Plot[kthContr[2,x],{x,1,20}]

Plot[kthContr[1,x]/x^.5,{x,1,20}]

Plot[{Sum[MangoldtLambda[n],{n,1,x}]
-
(x-Log[2Pi])
,
-kthContr[1,x]
}
,{x,1.2,20}]

Plot[{Sum[MangoldtLambda[n],{n,1,x}]
-
(x-Log[2Pi])
,
-kthContr[1,x]
-kthContr[2,x]
}
,{x,1.2,20}]

Plot[{Sum[MangoldtLambda[n],{n,1,x}]
-
(x-Log[2Pi])
,
-kthContr[1,x]
-kthContr[2,x]
-kthContr[3,x]
}
,{x,1.2,20}]

Plot[{Sum[MangoldtLambda[n],{n,1,x}]
-
(x-Log[2Pi])
,
-Sum[kthContr[k,x],{k,1,30}]
}
,{x,1.2,20}]

Plot[{Sum[MoebiusMu[n],{n,1,x}],Sqrt[x],-Sqrt[x]},{x,1,10000}]