Set $L=L(\chi_{-3},2)$ (related to your Gieseking constant) and $P=2\pi^2/81$. The limits are almost certainly (not proved) $$(-L/2,-P-L/2,-3P-L/2,\infty,3P-L/2,P-L/2)$$ Remark: I use a powerful extrapolation method explained for instance in a recent book of mine with K. Belabas, and in a few seconds obtain the limits to 38 decimals, which allowed me to guess the limits.

Set $Q=(1/2)L(\chi_{-3},2)$ (related to your Gieseking constant) and $P=2\pi^2/81$. The limits are almost certainly (not proved),

\begin{align} \lim_{m\to\infty}C_2(6m+0) &= -Q\\ \lim_{m\to\infty}C_2(6m+1) &= -P-Q\\ \lim_{m\to\infty}C_2(6m+2) &= -3P - Q\\ \lim_{m\to\infty}C_2(6m+3) &= \infty\\ \lim_{m\to\infty}C_2(6m-2) &= 3P - Q\\ \lim_{m\to\infty}C_2(6m-1) &= P - Q \end{align}

Remark: I use a powerful extrapolation method explained for instance in a recent book of mine with K. Belabas, and in a few seconds obtain the limits to 38 decimals, which allowed me to guess the limits.