Is this what you are after? Your problem was in the dollar signs around 3.3. You don't need to do add dollar signs when already inside a math-environment. Besides, 3.3 can be written both in normal text mode and in math.
Output

Code
\documentclass[11pt]{article}
\usepackage{fontspec}
\usepackage{siunitx}
\usepackage{mathtools}
\begin{document}
\begin{equation}
\phantom{0}LOD = 3.3 \cdot \frac{\phantom{0}S}{N} = 0.1\phantom{0} \si{\ng\per\uL}
\end{equation}
\end{document}
Suggestions
- Don't use
eqnarray. It has lots of problems. See The TUGboat-article by Lars Madsen. Use mathtools, which loads amstools, both of which has some really great functions for writing math. Especially align-environment is really useful. For single line equations, use equations-environment.
- Use
siunitx for getting consistent spacing and notation. I like to write all numerals with it, it makes it so much easier to change the look of everything.
- You don't need all those
\phantom-stuff, most likely.
- When writing words or abbreviations in math, use
\mathrm, otherwise, LOD would mathematically read out as L · O · D. See Which command should I use for textual subscripts in math mode?
Suggested Output

Suggested Code
\documentclass[11pt]{article}
\usepackage{fontspec}
\usepackage{siunitx}
\usepackage{mathtools}
\sisetup{per-mode=symbol} % or fraction, among others
\begin{document}
\begin{equation}
LOD = \num{3.3} \cdot \frac{S}{N} = \SI{0.1}{\ng\per\uL}
\end{equation}
\end{document}
eqnarray. Seeamsmathandmathtools. Thealign-environment is especially useful. – Runar Jun 23 '16 at 10:403.3\frac{...}{...}: what's the problem? – egreg Jun 23 '16 at 11:01\phantom{0}directives? – Mico Jun 23 '16 at 11:31