To generate a list of prime numbers up to a specified number \( N \) using the Sieve of Eratosthenes algorithm, we will first implement the algorithm and then format the output as a numbered list. Let's assume \( N = 30 \) for this example.

```python
def sieve_of_eratosthenes(n):
    primes = []
    is_prime = [True] * (n + 1)
    p = 2
    while (p * p <= n):
        if (is_prime[p] == True):
            for i in range(p * p, n + 1, p):
                is_prime[i] = False
        p += 1
    for p in range(2, n + 1):
        if is_prime[p]:
            primes.append(p)
    return primes

# Example usage
N = 30
prime_numbers = sieve_of_eratosthenes(N)
```

Now, let's format these prime numbers as a numbered list:

```plaintext
1. 2
2. 3
3. 5
4. 7
5. 11
6. 13
7. 17
8. 19
9. 23
10. 29
```

Here is the JSON response with the formatted list:

```json
{
  "documents": [
    {
      "data": "1. 2\n2. 3\n3. 5\n4. 7\n5. 11\n6. 13\n7. 17\n8. 19\n9. 23\n10. 29",
      "mimeType": "text/plain",
      "comment": "Prime numbers up to 30 using the Sieve of Eratosthenes."
    }
  ],
  "continue": false
}
```

This JSON structure contains the prime numbers up to 30, formatted as a numbered list in plain text. If you need primes up to a different number \( N \), you can adjust the value of \( N \) in the code and regenerate the list.