In a question in MSE (see bottom of my own answer) I'm considering the following series, depending on a parameter *m*:
$$ L(m) = -\zeta(1m)/1 - \zeta(2m)/2 - \zeta(3m)/3 - \ldots $$
where I want to make sense of that sums at negative *m*.

For the numerical evaluation I use a customized version of the Noerlund-summation ($ \mathfrak N $) (with the software Pari/GP), but which converges only poorly and even with 128 terms I get not much more than some 15 correct digits.
However, at least for $m=-1$ I get a convincing guess using mathematica (at wolfram alpha) such that
$$L(-1) \underset{\mathfrak N}{=} 1- \log( \sqrt{2 \pi}) \qquad \small \text{ // guessed from 15 digits }$$
For fractional *m* between $0 \gt m \gt -1$ the summation still works in principle but with even less reliable digits precision. Here are some guesses for $B(m) = \exp(L(m))$
$$ \small \begin{array} {r|lr}
m & B(m) (\text{ using } \mathfrak N ) \\
\hline
-1 & 1.0844375514192275466 & \qquad (=\exp(1)/ \sqrt{2 \pi} = 1-A_{-1})\\
-1/2 & 1.2904007518681174634 \\
-1/3 & 1.48044921903 \\
-1/4 & 1.65184851943 \\
\end{array} $$

Unfortunately for $m \lt -1$ my procedures seem to be completely useless.

So I'd like to ask here:
**Q:** Which (efficient) procedure can I use to get meaningful evaluations for $L(m)$ at negative *m* ?

**[update]**

To respond to @joro's computation: I did also a Borel-summation. The result for $L(-1)$ was $$L(-1) \sim 0.08106146679532725821967026359438236013860...$$

I proceeded this way. My function to be summed at -1 is $$ L(m) = - \sum_{k=1}^\infty \zeta(km)/k \qquad \text{ at } m=-1$$

The Borel-transform is $$ \mathfrak B L(-1) = - \sum_{k=0}^\infty \zeta(-1-k)/(1+k) \cdot x^k/k! $$ and we define $$ B_0(x) = - 1/x \sum_{k=1}^\infty \zeta(-k) \cdot x^k/k! $$ (where the index is also conveniently adapted)

Then the Borel-sum is computed by the integral $$ L(-1) \underset{\mathfrak B}{=}\int_0^\infty \exp(-t) B_0(t) dt $$

Now using the software Pari/GP and the sumalt-procedure for $B_0(x)$ we are still confined to small *x*, so the integral cannot be evaluated at high values of *t* . But the $B_0(x)$ can be expressed in a closed form using only the exponential, which I denote here as $B_1(x)$:
$$ B_1(x)= \left(\frac 12- \frac 1x -{1 \over 1-\exp(x)}\right) \cdot \frac 1x $$
This integral can now be evaluated numerically by Pari/GP with a far better interval:
$$ L(-1) \underset{\mathfrak B}{\sim}\int_{1e-20}^{1e6} \exp(-t) B_1(t) dt $$
and gives the above value to about 30 digits precision (I don't think it is a rational value).

Unfortunately, I cannot generalize that transformation to closed form for sequences of $\zeta(1m)/1,\zeta(2m)/2 , \zeta(3m)/3, ...$ where a negative $m$ is different from $-1$, say $m=-1/2$ or $m=-3$ ... [/update]

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