Are the roots of the Riemann zeta function approximated through numerical algorithms?

Question

Out of curiosity, I was taking a look at the R {pracma} package zeta() function, and specifically the example

## First zero on the critical line s = 0.5 + i t example in the documentation:

x <- seq(0, 20, len=1001)
z <- 0.5 + x*1i
fr <- Re(zeta(z))
fi <- Im(zeta(z))
fa <- abs(zeta(z))

the first root corresponds to

min(fa)  0.00418561897356751

which happens to be the absolute value of the zeta function at $0.5\,+\,14.14\,i$:

x[which.min(fa)] 14.14

However, $\small 0.00418561897356751$ is not too close an approximation to $\small 0$. For instance Wolfram alpha approximates the first non-trivial zero to $0.5 + 14.1347\,i$:

$$\tiny 0.5 + 14.13472514173469379045725198356247027078425711569924317568556746014996342980925676494901039317\,i$$

Questions:

[After the helpful comments received, it is clear that the roots of the Riemann zeta function are likely approximated numerically].

1. What degree of precision is needed to claim one additional root in the trillions of roots already found in the critical line?

2. Is there a way of using the zeta() function to look for roots of the zeta function? Are there other, perhaps more sanctioned (?) computer algorithms?

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NOTE: I am aware of the note on the linked page: The zeros are accurate to an absolute precision of $\pm 2\times 10^{−102}$, but I don't know its source.