The problem prompting this question is the following from Koblitz's Introduction to Elliptic Curves and Modular Forms 2nd ed, 1.6.1 (paraphrased):
Show the sum $\sum_{l\in \mathbb{Z}[i], \textrm{ }l\neq 0} \frac{1}{l^4}$ is a nonzero real number.
I can show the sum is a real number, but am stumped on showing it is nonzero. I can bound the sum, but this bound is not helpful. Since the sum is absolutely convergent, we can rearrange the terms in the series in the following way: \begin{align}\sum_{l\in \mathbb{Z}[i], \textrm{ }l\neq 0} \frac{1}{l^4} = 4\sum_{b=1}^\infty \frac{1}{b^4} + 4\sum_{a>0}\sum_{b>0} \frac{1}{(a+bi)^4} = 4\left(\zeta(4)+ \sum_{a>0}\sum_{b>0} \frac{1}{(a+bi)^4}\right). \end{align}
I got the right hand side by "rationalizing" the denominator to get $\frac{a^4+b^4-6a^2+b^2 +i(4ab^3-4a^3b)}{(a^2+b^2)^4}$, since the real part depends only on squares of $a,b$ we can sum over just the first quadrant and simply multiply this sum by 4. Perhaps this is the wrong approach, and there is an easier way.
More generally, I am interested in learning more about techniques which can be used to either evaluate sums over lattices or bound them (particularly from below). A resource with some worked out examples would be great, as well as some tricks you have found useful. An analogue for such a trick (for a single sum) might be something like GNUSupporter 8964民主女神 地下教會's answer to the question Prove $\lim_{m\to\infty}\sum_{k=1}^m\frac{2^{-k}}{k} = \log 2$.
While stackexchange has some questions about sums over lattices, most seem to be questions about convergence and not about techniques for evaluating them more generally.