Sure, I can create a test case generator for the "Life Game" problem. The generator will produce random valid inputs within the specified constraints. Here's the Python code for the test case generator:

```python
import random

def generate_initial_state(N, M):
    initial_state = []
    for i in range(2 * N - 1):
        row_length = N + abs(N - 1 - i)
        row = [random.randint(0, M - 1) for _ in range(row_length)]
        initial_state.append(row)
    return initial_state

def tcgen():
    N = random.randint(1, 6)
    M = random.randint(2, 10**9)
    L = random.randint(1, 10**9)
    
    initial_state = generate_initial_state(N, M)
    
    return N, M, L, initial_state

def print_test_case(N, M, L, initial_state):
    print(f"{N} {M} {L}")
    for row in initial_state:
        print(" ".join(map(str, row)))

# Example usage:
if __name__ == "__main__":
    # Generate a single test case
    N, M, L, initial_state = tcgen()
    print_test_case(N, M, L, initial_state)
    
    # Generate multiple test cases
    for i in range(5):  # Change the range as needed for more test cases
        N, M, L, initial_state = tcgen()
        print(f"Test Case {i + 1}:")
        print_test_case(N, M, L, initial_state)
        print()
    # Print end of input
    print("0 0 0")
```

This code does the following:

1. Defines a function `generate_initial_state` to generate the initial state of the hexagonal grid based on the given number of colonies on each edge (N) and the maximum number of viruses (M).
2. Defines the main `tcgen` function to generate random values for N, M, and L within the specified constraints.
3. Uses the `print_test_case` function to format and print the generated test case.
4. Includes an example usage block that generates and prints multiple test cases.

You can run this generator to produce different test cases for the "Life Game" problem, and it will ensure that all constraints are respected.