Series around $s=1$ for $F(s)=\int_{1}^{\infty}\text{Li}(x)\,x^{-s-1}\,dx$ 
Consider the function $$F(s)=\int_{1}^{\infty}\frac{\text{Li}(x)}{x^{s+1}}dx$$ where $\text{Li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral.  What is the series expansion around $s=1$?

It has a logarithmic singularity at $s=1$, and I am fairly certain (although I cannot prove it) that it should expand as something of the form $$\log (1-s)+\sum_{n=0}^\infty a_n (s-1)^n.$$  (An expansion of the above form is what I am looking for)  I also have a guess that the constant term is $\pm \gamma$ where $\gamma$ is Euler's constant.  Does anyone know a concrete way to work out such an expansion?
Thanks!

By looking at the sum $$\sum_{k=2}^N\frac{1}{k\log k}=\log \log N+K+O\left(\frac{1}{N}\right)$$ in this question, I found another way to prove that as $s\rightarrow 0$ $$\int_2^\infty \frac{x^{-s-1}}{\log x}dx=-\log(s)-\gamma-\log \log 2+O(s\log(s)).$$ Let $\Lambda(n)$ be the Von Mangoldt Lambda function, and $\gamma_0$ the Euler-Mascheroni Constant. Then we have the expansion of the similar sum $$\sum_{n\leq x}\frac{\Lambda(n)}{n\log n}=\log\log x+\gamma_{0}+O\left(\frac{1}{\log x}\right),$$ which appears in the proof of theorem 2.7 in Montgomery and Vaughn. Let $$S(x)=\sum_{2\leq k\leq x}\frac{1}{k\log k}- \sum_{n\leq x}\frac{\Lambda(n)}{n\log n},$$ and examine $I=\delta \int_1^\infty S(x)x^{-\delta -1}dx$ as $\delta\rightarrow 0$. As $S(x)=(K-\gamma_0)+O(1/\log x)$, it follows that $I=K-\gamma_0+O(\delta \log (1/\delta)$. Then, since $$\sum_{n=1}^{\infty}a_{n}n^{-s}=s\int_{1}^{\infty}A(x)x^{-s-1}dx$$ (Theorem 1.3 of Montgomery and Vaughn) we see that $$\sum_{n=2}^{\infty}\frac{n^{-\delta}}{n\log n}-\log \zeta(\delta+1)=O(\delta\log(1/\delta)$$ as $\delta\rightarrow 0$, and so $$\sum_{n=2}^\infty \frac{n^{-\delta-1}}{\log n}=-\log \delta+(K-\gamma)+O(\delta\log(1/\delta).$$ Now, writing the left hand side as a Riemann Stieltjes integral and simplifying the expression for $K$ combined with the resulting terms allows us to conclude that $$\int_{2}^\infty \frac{x^{-\delta-1}}{\log x}dx=-\log \delta -\gamma-\log \log 2+O\left(\delta\log(1/\delta)\right).$$
