Step 1: Concept
For a uniform object, the center of mass is located at its geometric center.
Step 2: Meaning
The rod is uniform and has length L.
Step 3: Analysis
The geometric center of a rod starting from $x=0$ to $x=L$ is the midpoint.
Step 4: Conclusion
The midpoint is $\frac{L}{2}$.
Final Answer: (C)