Difficult interpolation problem

Let:

r(1) = \sum_{n=1}^{\infty} \frac{1}{n^n}

r(2) = \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{(mn)^{mn}}

r(3) =\sum_{l=1}^{\infty} \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \frac{1}{(lmn)^{lmn}}

Find the most natural interpolation of r(z) on the complex plane \mathbb{C} given the above scheme.

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