Riemann Prime Counting Function correction/pairing terms by Mathematica

Riemann Prime Counting Function:

$$f(x)=operatorname{li}(x)-sum_rhooperatorname{li}(x^rho)-ln 2+int_x^infty frac{mathrm dt}{t(t^2-1)ln t}$$

The second correction/paring terms:

$$sum_rhooperatorname{li}(x^rho)=sum_{I[rho]>0}[operatorname{Li}(x^rho)+operatorname{Li}(x^{1-rho})]$$

I tried to use Mathematica function LogIntegral to plot this second correction/paring terms, for example, when I only include the first 2 non-trivial zeros, and plot with range x from 1 to 5:

Plot[Sum[LogIntegral[x^ZetaZero[k]] + LogIntegral[x^(1 - ZetaZero[k])],
     {k, 1, 2}], {x, 1, 5}]

However, I got very large value instead of small correction:

I can also use simplified equation provided by reference 1:

$$operatorname{li}(x^rho)=operatorname{li}(e^{rho log x})sim frac{e^{rho log x}}{rho log x}$$

Plot[Sum[Exp[ZetaZero[k]*Log[x]]/(ZetaZero[k]*Log[x]), {k, 1, 2}] +
     Sum[Exp[(1 -ZetaZero[k])*Log[x]]/((1 - ZetaZero[k])*Log[x]),
         {k, 1, 2}], {x, 1, 5}]

Then I got the correct result:

Anyone knows what is wrong for the LogIntegral one?

Thank you!

1: H. Riesel and G Gohl, "Some Calculations Related to Riemann’s Prime Number Formula," Mathematics of Computation, 24(112), 1970 pp. 969–983.

Plot[
     Sum[
         ExpIntegralEi[ZetaZero[k]*Log[x]] + 
         ExpIntegralEi[(1 - ZetaZero[k])*Log[x]],
     {k, 1, 2}], {x, 1, 5},
Frame -> True]