We present deep GMOS long-slit spectroscopy of 15 Coma cluster S0 galaxies, and extract kinematic properties along the major axis to several times the disc scale-length. Supplementing our dataset with previously published data, we create a combined sample of 29 Coma S0s, as well as a comparison sample of 38 Coma spirals. Using photometry from SDSS and 2MASS, we construct the Tully-Fisher relation (TFR; luminosity versus maximum rotational velocity) for S0 galaxies. At fixed rotational velocity, the Coma S0 galaxies are on average fainter than Coma spirals by 1.10$pm$0.18, 0.86$pm$0.19 and 0.83$pm$0.19 mag in the g, i and Ks bands respectively. The typical S0 offsets remain unchanged when calculated relative to large field-galaxy spiral samples. The observed offsets are consistent with a simple star formation model in which S0s are identical to spirals until abrupt quenching occurs at some intermediate redshift. The offsets form a continuous distribution tracing the time since the cessation of star formation, and exhibit a strong correlation (>6{sigma}) with residuals from the optical colour-magnitude relation. Typically, S0s which are fainter than average for their rotational velocity are also redder than average for their luminosity. The S0 TFR offset is also correlated with both the projected cluster-centric radius and the {Sigma} (projected) local density parameter. Since current local environment is correlated with time of accretion into the cluster, our results support a scenario in which transformation of spirals to S0s is triggered by cluster infall.
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