I was trying to get some interesting result for $\zeta(3)$, exploring the following function: $$W(a) = \sum_{k=1}^\infty \frac{1}{k^3 + a^3}, \mbox{ with } \lim_{a\rightarrow 0} W(a) = \zeta(3).$$

Let $w_1, w_2, w_3$ be the three roots (one real, two complex) of $(w+1)^3+a^3=0$, with $w_1=-(a+1)$. Also, $a$ is a real number. Using Wolfram Alpha (see here), I get

$$W(a)=\frac{-1}{3}\cdot\sum_{j=1}^3 W_j(a), \mbox{ with } W_j(a) = \frac{\psi^{(0)}(-w_j)}{(w_j+1)^2}.$$

Here $\psi^{(0)}$ is the digamma function. The result is wrong because $W_1(a) \rightarrow \infty$ as $a\rightarrow 0^+$ while $W_2(a)$ and $W_3(a)$ remain bounded. Indeed using $a=0.0001$, Wolfram yields $W(a)\approx -2334.16$, see here. Surprising, with $a=0.01$ it yields $W(a)\approx 1.20206$ which is very close to the true result.

Surprisingly, Wolfram knows (see here) that $$\lim_{a\rightarrow 0} W(a) = -\frac{\psi^{(2)}(1)}{2}.$$

Of course (this is a well known fact), $\zeta(3)=-\psi^{(2)}(1)/2$ and thus Wolfram is correct this time.

My question:

What is going on with this computation (or is it me?), and what is the correct formula for $W(a)$?