On the internet we could find the Dirichlet L-series https://www.lmfdb.org/L/1/3/3.2/r1/0/0
$$ L(s, \chi_3) = 1 - 2^{-s} +4^{-s}-5^{-s}+7^{-s}-8^{-s}+10^{-s}-11^{-s}+13^{-s}-14^{-s}+16^{-s}-17^{-s}+\dots $$
If I try to set the value to $s=0$ or $s=-1$ we obtain:
- $L(0, \chi_3) = (1 -1 + 0) + (1 - 1 + 0) + (1-1+0)+ \dots$
- $L(-1, \chi_3) = 1 -2 + 0 + 4 - 5 + 0 + 6 - 7 + 0 + \dots $
The web page tells us the value at $s = 1$:
$$ L(1, \chi_3) = 0.6045997880 $$
There is a functional equation listed as:
$$ \Lambda(s) = 3^{s/2}\Gamma(s+1)L(s) = \Lambda(1-s) $$
If we set $s = 0$, we get a plausible looking value for the divergent series just written.
$$ 3^{0/2}\Gamma(0+1)L(0, \chi_3) = 3^{1/2}\Gamma(2)L(1, \chi_3) $$
For reference, the Dirichlet character is written as a table:
$$ \begin{array}{c|rr} & 1 & 2 \\ \hline \chi_1 & 1 & 1 \\ \chi_2 & 1 & -1 \end{array}$$