Consider the set $\mathcal{T}$ of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let $\triangle \in \mathcal{T}$ be the triangle with least perimeter. If $a^{\circ}$ is the largest angle of $\triangle$ and $L$ is its perimeter, determine the value of $\frac{a}{L}$.
1. **Identify the triangle with the least perimeter:**
   - We need to find a triangle with sides that are distinct prime numbers in arithmetic progression.
   - The smallest such set of primes is \(3, 5, 7\).
   - Therefore, the triangle with sides \(3, 5, 7\) has the least perimeter.

2. **Calculate the perimeter \(L\):**
   \[
   L = 3 + 5 + 7 = 15
   \]

3. **Determine the largest angle \(a\):**
   - The largest angle is opposite the longest side, which is \(7\).
   - Use the Law of Cosines to find the angle opposite the side of length \(7\):
     \[
     c^2 = a^2 + b^2 - 2ab \cos(C)
     \]
     Here, \(a = 3\), \(b = 5\), and \(c = 7\). We need to find \(\cos(C)\):
     \[
     7^2 = 3^2 + 5^2 - 2 \cdot 3 \cdot 5 \cdot \cos(C)
     \]
     \[
     49 = 9 + 25 - 30 \cos(C)
     \]
     \[
     49 = 34 - 30 \cos(C)
     \]
     \[
     30 \cos(C) = 34 - 49
     \]
     \[
     30 \cos(C) = -15
     \]
     \[
     \cos(C) = -\frac{1}{2}
     \]

4. **Find the angle \(C\):**
   - Since \(\cos(C) = -\frac{1}{2}\), the angle \(C\) is:
     \[
     C = 120^\circ
     \]

5. **Calculate \(\frac{a}{L}\):**
   - Here, \(a = 120\) and \(L = 15\):
     \[
     \frac{a}{L} = \frac{120}{15} = 8
     \]

The final answer is \(\boxed{8}\).