There are about twenty values physics have found that appear to be so finely tuned to enable complex life to exist.
A few of them are:
The cosmological constant – tuned to 120 decimal places – one part in a trillion trillion trillion trillion trillion trillion trillion trillion trillion trillion, or the universe would be such that it would be impossible for us to evolve.
1 second after the big bang the expansion rate and the mass density have to be adjusted to each other just right; if the universe were expanding just one part in the fifteenth decimal place. One degree faster, and the universe would have flown apart, without ever forming galaxies. One value slower, at the fifteenth decimal place, and the universe would have expanded to its maximum size and then collapsed, without us ever having reached this point in our existence.

What are the arguments against fine-tuning? Well, positing a multiverse in which tunings  of values for gravity, mass, the cosmological constant, and other values are randomly ascribed allows pure chance to dictate that our universe happens to be the one in which life can exist. This is the only way out for a pure materialist, but there are a few athiestic scientists who go against the opinion of most thinkers (whether athiest or not) and disagree that there is such a thing as fine tuning at all. See this Luke Barnes review of Victor Stenger’s critique of cosmic fine-tuning. Here is a quote excerpted from this post:

Finally, and most importantly, note carefully Stenger’s conclusion. He states that no fine-tuning is needed for the neutron-proton mass difference in our universe to be approximately equal to the up quark-down quark mass difference in our universe. Stenger has compared our universe with our universe and found no evidence of fine-tuning. There is no discussion of the life-permitting range, no discussion of the possible range of [mass(neutron) – mass(proton)] (or its relation to the possible range of [mass(down quark) – mass(up quark)], and thus no relevance to fine-tuning whatsoever. (p. 51)

For a closer look at the origin of the cosmological constant, check out this video which describes how Einstein came up with it, then turned away from it, and later, experiments demonstrated that it indeed seems valid.