The $n$-th Gram point can be approximated with the formula:
$$\frac{2 \pi \left(n-\frac{7}{8}\right)}{W\left(\frac{n-\frac{7}{8}}{\exp (1)}\right)}$$
and the $n$-th Franca-Leclair point can be approximated with the formula:
$$\frac{2 \pi \left(n-\frac{11}{8}\right)}{W\left(\frac{n-\frac{11}{8}}{\exp (1)}\right)}$$
where $W$ is the Lambert-W function.
Gram points are points $t$ such that:
$$\Re\left(\zeta \left(i t+\frac{1}{2}\right)\right) \neq 0$$ and $$\Im\left(\zeta \left(i t+\frac{1}{2}\right)\right) = 0$$
Franca-Leclair points are points $t$ such that:
$$\Re\left(\zeta \left(i t+\frac{1}{2}\right)\right) = 0$$ and $$\Im\left(\zeta \left(i t+\frac{1}{2}\right)\right) \neq 0$$
where $\zeta(s)$ is the Riemann zeta function.
The maximum number of zeros in between a Gram point and Franca Leclair point appears to be 2, and as an example I use this plot of the real and imaginary part of the Riemann zeta function which shows 2 consecutive zeta zeros with no Gram point or Franca-Leclair point in between them. It is also an example of the Lehmer's phenomenon:
Mathematica code for the plot:
Show[Plot[Re[Zeta[1/2 + I*t]], {t, 17143.8 - 1/20, 17143.8 + 1/20},
PlotRange -> {-0.005, 0.005}],
Plot[Im[Zeta[1/2 + I*t]], {t, 17143.8 - 1/20, 17143.8 + 1/20},
PlotRange -> {-0.005, 0.005}]]
My question is:
Can there be more than 2 zeta zeros in between a Gram point and a Franca-Leclair point?