I found a new sum for
\text\left[\text{PolyLog}\left[3,\frac{1}{a}\right]\right]
The real part is casi equal althought must be equal the imagimary part is equal for 0<a <1 it is possible that the error machine afect the result somebody can check it .
\frac{1}{a}\left(1+\left(\left(\frac{1}{2} \left(\frac{1}{2} \left(1-\text{Log}[a]+a \text{Gamma}[0,\text{Log}[a]] \text{Log}[a]^2\right)+\frac{1-2 \text{Log}[a]+4 a^2 \text{Log}[a]^2 (-\text{CoshIntegral}[2 \text{Log}[a]]+\text{SinhIntegral}[2 \text{Log}[a]])}{8 a}\right)+\left(\frac{1}{48} \left(-39 a+36 a \text{Log}[a]-4 a \pi ^2 \text{Log}[a]-24 a \text{Log}[a]^2+24 a \text{Log}[2] \text{Log}[a]^2+36 a \text{Zeta}[3]\right)+\frac{1}{4} a \text{Log}[2] \text{Log}[a]^2\right)\right)\right)\right)
Thanks