Here's an example, but you may consider it to be a bit artificial Consider the following language: s is in $L$ if $S=(x,y)$ where $x$ is a 3-SAT instance, $y$ is a satisfying instance for $y$, and also satisfying that if $n$ is the number of true variables in $y$, then the $n$th non-trivial zero of the zeta function has real part equal to $1/2$. But this is a bit artificial. My guess is that you specified your five examples in order to avoid that.
Edit: The following does not work. Possibly it can be fixed, but see Emil's comment below.
But we can modify this is a little bit to cheat and get an example of 1, provided one is willing to use GRH rather than RH; my guess is that a similar construction can be done with RH, but it doesn't immediately jump out at me. The first detail is that assuming GRH, the smallest positive primitive root $u$, of a prime $p$ is bounded above by a power of $\log p$. In particular, Shoup showed that $u = O( (\log p)^6)$.
Given positive constants $c_1$, $c_2$, and a function $f(n)$ Let $A_{c_1,c_2, f(n)}$ be the language given by the following: $a$ is in $A$ if $a=(x,y,z,w)$ where $x$, $y$, and $z$ are defined as follows. We have $x$ is a 3-SAT instance and $y$ is a satisfying set of variables for $x$. $z$ is a prime $p$ such that $y< p < (c_1)p^{c_2}$ (where we are thinking of our list $y$ of variable assignments as just a binary number). We have $w$ is a string of 1s where the number of 1s is a primitive root of $p$, and $w$'s length is bounded by $f(p)$.
Now, we can choose a $c_1$ and $c_2$ such that there is always such a prime $p$ when there is a $y$. (Baker, Harman and Pintz showed that one can take $c_2 = 0.525$). So choose such a $c_1$ and $c_2$. Then, note that if we believe GRH, we can always find such a $w$ which is polynomial in the length of $x$, and we can choose $f(n)$ so that $w$'s length is guaranteed to be less than $f(p)$ if GRH is true.
So now our NP-hard question then is just is a given $x$ a valid $x$ for an $(x,y,z,w)$ in $A$? If GRH is true, we can always find a $y$, $z$ and $w$ which is guaranteed to be just polynomial in length when $x$ is true. But without GRH we can only get that this is NP-hard since our $w$ may be too large.
This is essentially just a padding trick. My guess is that a similar padding trick can be used with just RH as an assumption but I don't immediately see how to do it. And this is still a pretty artificial problem. I'd be interested to see if someone has a more natural example.