"no" can be an adverb or an adjective. "without" is a preposition, so "no" and "without" are very different words and don't have any relation between themselves. The only similarity between "no" and "without" is that "without" is the negative form of "with", so both words are used to express a negation.
You can replace "without" by "no" if you reconstruct the phrase so that the phrase doesn't need a preposition anymore. So, your phrase "My investigations were without result" can be replaced by "My investigations had no results." because this second phrase doesn't require a preposition, this doesn't mean that "no" and "without" are synonyms, they are completely different words because one is a preposition and the other is an adjective in this example.