It doesn't. You have no solid evidence for or against God, again, depending on definition.
Unfortunately logic doesn't work that way due to the extreme difficulty of proving a negative. You start at the negative position and prove the positive position.
Feel free to try and work it back the other way, but bear in mind that in doing so you are acting illogically, and will be treated as such.
Logic works exactly like that. You give premises and based on them, you can figure out if something is true, and evidence or hypothesis based on evidence usually work to build premises on. In science. You do not believe in things unless they are rationally backed up by empirical evidence, as some things are extremely hard to disprove. Science gives you information you can use for thinking, creating stuff, whatever. But it doesn't, and never will, cover anything that cannot be empirically tested and falsified. Dropping ball from 10 feet doesn't cause "Law of God" to apply, cause usually we just rather call it gravity. And if thunder would strike ball every time it is dropped from 10 feet, we would have some kind of theory explaining why it happens, or not. Yet God cannot be accepted as explanation, because it's not fallible theory. It might be the best one, but it's still not objectively any better than other similar theories.
In philosophy, however, you can base things on evidence, but you can also base something simply on thought. Like atomism. You couldn't have based the idea on evidence, it was just based on though about what the world is like. Rational thought of rational person. "Hey, maybe everything is built out of blocks?" It would've sounded only a bit more crazy to materialists than some higher being, but turns out it was true.
I don't believe 3 billion people can be any more right than one, I've seen enough of idiotic belief in authority to think it isn't true. I also do not think matter or laws of physics need a creator, I've no problem believing cyclic or just infinite universe is possible. Thus, for me, rational thing is not to believe in God. I don't need God for my world view to work, nor I do need it for my mental health. I however cannot impose this on other, as I cannot expect everyone to see world as materialistic, reductionist and deterministic, with ethics based on same thing I base ethics on. These are philosophical questions everyone has their own answers to, or doesn't, and it builds up the subjective universe for them that really doesn't matter to anyone else.
Saying God doesn't exist is kinda like saying Ethical non-naturalism is wrong. I can say I do not believe Ethical non-naturalism is true, I cannot say it's not true. Well, I can, but then I also assume only my world view is correct.