I propose that you should keep it at 4 degree, and for long term, lenti should be stock at -80C.

Maybe you should freeze them as viable cells. This suggestion is for long term storage. Note pH fluctuations in short-term experiments.

Good Luck

Usually, virus would be stable at 4 degree for short term storage.  For very long storage, I would suggest -80 degree, while alloquited. FBS is sometimes added for the protection against degradation.

Though virus is stored for a day, it should be in small quantity with multiple vials and than store at 4 degrees ( Only for a day ).

Is this supernatant from cells cultured for the production of virus for downstream assays? If so, I would recommend storing a large quantity of aliquots at -80 ºC. Thaw one of the frozen stocks for titration and thaw the rest as needed. I would not suggest storing retroviruses at anything other than -80 or on ice for minutes before and after experiments. 

I recommend to use multiple vials at lest 3 and storage at -20C° to be in safe side. Then thawed ones for working assay. 

For short time (few hours) you can put at +4 ºC and for long storage you keep at -81 ºC but don't forget make as many aliquots as possible to reduce the shock of thawing and freezing.

I suggest that it is not advisable, For a short time you can keep at 40C before use it for the experiment. Otherwise it has to be at -80.  At  40C it looses  its titer.  Every time freeze thaw is also not advisable. So we have to make approximate required amount vials for each experiment, depending upon its titer.

It may be placed for short duration for short time but long storage needs -80 C but this short duration also varies from virus to virus as I am working on IBDV it bears short interval

For short term storage I prefer 4 centigrade degree. -20 centigrade degree may affect the content of supernatant due to freezing thawibg.

Keep it at 4 degrees maximum overnight and store in aliquotes at -80C to avoid thawing and refreezing

Ideally, viral particles can be stored at 4 degree C for couple of days, but prolonged storage at this temperature significantly looses the viral titre.

For long term storage, the viral supernatant can be stored at -80 degree, the FBS in culture media can act as cryoprotectant or you can add some FBS. Store these in different aliquot to prevent repetative freeze-thaw.