We present the discovery and characterization of five hot and warm Jupiters -- TOI-628 b (TIC 281408474; HD 288842), TOI-640 b (TIC 147977348), TOI-1333 b (TIC 395171208, BD+47 3521A), TOI-1478 b (TIC 409794137), and TOI-1601 b (TIC 139375960) -- based on data from NASAs Transiting Exoplanet Survey Satellite (TESS). The five planets were identified from the full frame images and were confirmed through a series of photometric and spectroscopic follow-up observations by the $TESS$ Follow-up Observing Program (TFOP) Working Group. The planets are all Jovian size (R$_{rm P}$ = 1.01-1.77 R$_{rm J}$) and have masses that range from 0.85 to 6.33 M$_{rm J}$. The host stars of these systems have F and G spectral types (5595 $le$ T$_{rm eff}$ $le$ 6460 K) and are all relatively bright (9 $<V<$ 10.8, 8.2 $<K<$ 9.3) making them well-suited for future detailed characterization efforts. Three of the systems in our sample (TOI-640 b, TOI-1333 b, and TOI-1601 b) orbit subgiant host stars (log g$_*$ $<$4.1). TOI-640 b is one of only three known hot Jupiters to have a highly inflated radius (R$_{rm P}$ > 1.7R$_{rm J}$, possibly a result of its host stars evolution) and resides on an orbit with a period longer than 5 days. TOI-628 b is the most massive hot Jupiter discovered to date by $TESS$ with a measured mass of $6.31^{+0.28}_{-0.30}$ M$_{rm J}$ and a statistically significant, non-zero orbital eccentricity of e = $0.074^{+0.021}_{-0.022}$. This planet would not have had enough time to circularize through tidal forces from our analysis, suggesting that it might be remnant eccentricity from its migration. The longest period planet in this sample, TOI-1478 b (P = 10.18 days), is a warm Jupiter in a circular orbit around a near-Solar analogue. NASAs $TESS$ mission is continuing to increase the sample of well-characterized hot and warm Jupiters, complementing its primary mission goals.
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