Setting $$z=1/12$$ gives:
$$\mathscr{S}_{\infty}\left(\frac{1}{12}\right)=\sum_{k=1}^{\infty}\frac{\log k}{k^2}\cos\left(\frac{\pi k}{6}\right)=$$$$\cos\left(\frac{\pi}{6}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+1)}{(12k+1)^2}+
\cos\left(\frac{\pi}{3}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+2)}{(12k+2)^2}+$$
$$\cos\left(\frac{\pi}{2}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+3)}{(12k+3)^2}+
\cos\left(\frac{2\pi}{3}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+4)}{(12k+2)^4}+$$
$$\cos\left(\frac{5\pi}{6}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+5)}{(12k+5)^2}+
\cos\left(\pi \right)\, \sum_{k=0}^{\infty}\frac{\log(12k+6)}{(12k+6)^2}+$$
$$\cos\left(\frac{7\pi}{6}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+7)}{(12k+7)^2}+
\cos\left(\frac{4\pi}{3}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+8)}{(12k+8)^2}+$$
$$\cos\left(\frac{3\pi}{2}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+9)}{(12k+9)^2}+
\cos\left(\frac{5\pi}{3}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+10)}{(12k+10)^2}+$$
$$\cos\left(\frac{11\pi}{6}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+11)}{(12k+11)^2}+
\cos\left(2\pi\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+12)}{(12k+12)^2}$$Using $$\cos(\pi \pm \theta) = -\cos \theta$$ and $$\cos(2\pi - \theta) = \cos \theta$$, we express the second 6 cosines in terms of the first 6:
$$\cos\left(\frac{\pi}{6}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+1)}{(12k+1)^2}+
\cos\left(\frac{\pi}{3}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+2)}{(12k+2)^2}+$$
$$
-\cos\left(\frac{\pi}{3}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+4)}{(12k+4)^2}+$$
$$-\cos\left(\frac{\pi}{6}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+5)}{(12k+5)^2}- \sum_{k=0}^{\infty}\frac{\log(12k+6)}{(12k+6)^2}+$$
$$-\cos\left(\frac{\pi}{6}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+7)}{(12k+7)^2}-
\cos\left(\frac{\pi}{3}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+8)}{(12k+8)^2}+$$
$$
\cos\left(\frac{\pi}{3}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+10)}{(12k+10)^2}+$$
$$\cos\left(\frac{\pi}{6}\right)\, \sum_{k=0}^{\infty}\frac{\log(12k+11)}{(12k+11)^2}+
\sum_{k=0}^{\infty}\frac{\log(12k+12)}{(12k+12)^2}$$Combining series with equivalent Cosine coefficients, this becomes:$$\cos\left(\frac{\pi}{6}\right)\, \sum_{k=0}^{\infty} \Bigg\{
\frac{\log(12k+1)}{(12k+1)^2} -
\frac{\log(12k+5)}{(12k+5)^2} -
\frac{\log(12k+7)}{(12k+7)^2} +
\frac{\log(12k+11)}{(12k+11)^2}
\Bigg\} +$$$$\cos\left(\frac{\pi}{3}\right)\, \sum_{k=0}^{\infty} \Bigg\{
\frac{\log(12k+2)}{(12k+2)^2} -
\frac{\log(12k+4)}{(12k+4)^2} -
\frac{\log(12k+8)}{(12k+8)^2} +
\frac{\log(12k+10)}{(12k+10)^2}
\Bigg\} +$$$$- \sum_{k=0}^{\infty}\frac{\log(12k+6)}{(12k+6)^2}+ \sum_{k=0}^{\infty} \frac{\log(12k+12)}{(12k+12)^2}=$$
$$\frac{ \cos\left(\frac{\pi}{6}\right)}{144} \, \log12 \, \Bigg\{
\psi_1\left(\tfrac{1}{12}\right)
- \psi_1\left(\tfrac{5}{12}\right)
- \psi_1\left(\tfrac{7}{12}\right)
+\psi_1\left(\tfrac{11}{12}\right)
\Bigg\}$$$$-\frac{ \cos\left(\frac{\pi}{6}\right)}{144} \Bigg\{
\zeta ' \left(2, \tfrac{1}{12} \right)
- \zeta ' \left(2, \tfrac{5}{12} \right)
- \zeta ' \left(2, \tfrac{7}{12} \right)
+ \zeta ' \left(2, \tfrac{11}{12} \right)
\Bigg\}$$$$+ \frac{ \cos\left(\frac{\pi}{3}\right)}{144} \, \log12 \, \Bigg\{
\psi_1\left(\tfrac{1}{6}\right)
- \psi_1\left(\tfrac{1}{3}\right)
- \psi_1\left(\tfrac{2}{3}\right)
+\psi_1\left(\tfrac{5}{6}\right)
\Bigg\}$$$$-\frac{ \cos\left(\frac{\pi}{3}\right)}{144} \Bigg\{
\zeta ' \left(2, \tfrac{1}{6} \right)
- \zeta ' \left(2, \tfrac{1}{3} \right)
- \zeta ' \left(2, \tfrac{2}{3} \right)
+ \zeta ' \left(2, \tfrac{5}{6} \right)
\Bigg\}$$$$-\frac{(\eta(2)\, \log 6-\eta ' (2))}{36} $$By the reflection formula for the Trigamma function, all of those Trigammas can be reduced to squared cosecants:$$\frac{ \cos\left(\frac{\pi}{6}\right)}{144} \, \log12 \, \Bigg\{
\pi^2 \csc^2 \left(\frac{\pi}{12}\right)
- \pi^2 \csc^2 \left(\frac{5\pi}{12}\right)
\Bigg\} +$$$$\frac{ \cos\left(\frac{\pi}{3}\right)}{144} \, \log12 \, \Bigg\{
\pi^2 \csc^2 \left(\frac{\pi}{6}\right)
- \pi^2 \csc^2 \left(\frac{\pi}{3}\right)
\Bigg\}$$$$-\frac{ \cos\left(\frac{\pi}{6}\right)}{144} \Bigg\{
\zeta ' \left(2, \tfrac{1}{12} \right)
- \zeta ' \left(2, \tfrac{5}{12} \right)
- \zeta ' \left(2, \tfrac{7}{12} \right)
+ \zeta ' \left(2, \tfrac{11}{12} \right)
\Bigg\}$$$$-\frac{ \cos\left(\frac{\pi}{3}\right)}{144} \Bigg\{
\zeta ' \left(2, \tfrac{1}{6} \right)
- \zeta ' \left(2, \tfrac{1}{3} \right)
- \zeta ' \left(2, \tfrac{2}{3} \right)
+ \zeta ' \left(2, \tfrac{5}{6} \right)
\Bigg\}$$$$-\frac{(\eta(2)\, \log 6-\eta ' (2))}{36} $$Expressing all of the trig terms in surd form (see Sine and Cosine here:
Elementary Functions ), we get the following:
$$\frac{\pi^2}{48}\left(4+3\sqrt{6}\right) -\frac{(\eta(2)\, \log 6-\eta ' (2))}{36} $$$$-\frac{ 1}{ 96\sqrt{3} } \Bigg\{
\zeta ' \left(2, \tfrac{1}{12} \right)
- \zeta ' \left(2, \tfrac{5}{12} \right)
- \zeta ' \left(2, \tfrac{7}{12} \right)
+ \zeta ' \left(2, \tfrac{11}{12} \right)
\Bigg\}$$$$-\frac{ 1}{288} \Bigg\{
\zeta ' \left(2, \tfrac{1}{6} \right)
- \zeta ' \left(2, \tfrac{1}{3} \right)
- \zeta ' \left(2, \tfrac{2}{3} \right)
+ \zeta ' \left(2, \tfrac{5}{6} \right)
\Bigg\}$$