How do I do a search filtered by date AND time?

This works:

var dt = format.format({type: format.Type.DATE, value: new Date()});
search.create({ type: "supportcase", filters: [['lastmodifieddate', search.Operator.AFTER, dt]], columns: ["internalid"] }).run().getRange(0, 1000);

But this results in UNEXPECTED_ERROR:

var dt = format.format({type: format.Type.DATETIME, value: new Date()});
search.create({ type: "supportcase", filters: [['lastmodifieddate', search.Operator.AFTER, dt]], columns: ["internalid"] }).run().getRange(0, 1000);

format.Type.DATETIMETZ also causes UNEXPECTED_ERROR.

Date isn’t enough, I need to filter by date AND time. After several years with NS I shouldn’t be surprised at how difficult this is, but I am.