3.9 US quarts. At least on my 2007 VTX 1300C
The 2005 Honda VTX1300 uses a Pro Honda GN4 or a Pro Honda HP4 (without molybdenum additives) 4 stroke motorcycle oil, or it's equivalent. The GN4 being a conventional Motor Oil and the HP4 is a Semi-Synthetic blend.
Recommended Viscosity is SAE 10W-30, but 10W-40 is fine. The difference between the two would be the ambient temperature that they can be used in. For colder climates, use the 10W-30. For warmer climates, use the 10W-40.
The unit takes about 3.9 U.S. Quarts.
Useless Oil Information:
API Classification: SG or higher except oils labeled as energy conserving on the circular API service label.
JASO T 903 standard: MA
Do not use oil additives. Do not use oils with graphite or molydbenum additives.