Template to convert any lambda function (including capturing lambdas) to a std::function object

I have the following code that can convert a lambda into a C-style function pointer. This works for all lambdas including lambdas with captures.

#include <iostream>
#include <type_traits>
#include <utility>

template <typename Lambda>
struct lambda_traits : lambda_traits<decltype(&Lambda::operator())>
{};

template <typename Lambda, typename Return, typename... Args>
struct lambda_traits<Return(Lambda::*)(Args...)> : lambda_traits<Return(Lambda::*)(Args...) const>
{};

template <typename Lambda, typename Return, typename... Args>
struct lambda_traits<Return(Lambda::*)(Args...) const>
{
    using pointer = typename std::add_pointer<Return(Args...)>::type;

    static pointer to_pointer(Lambda&& lambda)
    {
        static Lambda static_lambda = std::forward<Lambda>(lambda);
        return [](Args... args){
            return static_lambda(std::forward<Args>(args)...);
        };
    }
};

template <typename Lambda>
inline typename lambda_traits<Lambda>::pointer to_pointer(Lambda&& lambda)
{
    return lambda_traits<Lambda>::to_pointer(std::forward<Lambda>(lambda));
}

This can be used as follows to pass a lambda with a capture into a C-style API:

// Function that takes a C-style function pointer as an argument
void call_function(void(*function)())
{
    (*function)();
}

int main()
{
    int x = 42;

    // Pass the lambda to the C-style API
    // This works even though the lambda captures 'x'!
    call_function(to_pointer([x] {
        std::cout << x << std::endl;
        }));
}

Given this, it seems like it should be relatively straightforward to write a similar template that can convert lambdas (including lambdas with captures) generically into std::function objects, but I am struggling to figure out how. (I am not super familiar with template meta-programming techniques so I am a bit lost)

This is what I tried, but it fails to compile:

template <typename Lambda>
struct lambda_traits : lambda_traits<decltype(&Lambda::operator())>
{};

template <typename Lambda, typename Return, typename... Args>
struct lambda_traits<typename std::function<Return(Args...)>> : lambda_traits<typename std::function<Return(Args...)> const>
{};

template <typename Lambda, typename Return, typename... Args>
struct lambda_traits<typename std::function<Return(Args...)> const>
{
    using pointer = typename std::function<Return(Args...)>*;

    static pointer to_pointer(Lambda&& lambda)
    {
        static Lambda static_lambda = std::forward<Lambda>(lambda);
        return [](Args... args) {
            return static_lambda(std::forward<Args>(args)...);
        };
    }
};

template <typename Lambda>
inline typename lambda_traits<Lambda>::pointer to_pointer(Lambda&& lambda)
{
    return lambda_traits<Lambda>::to_pointer(std::forward<Lambda>(lambda));
}

This fails to compile and says that the Lambda template parameter is not being used by the partial specialization.

What is the correct way to do this?

(Note, I am stuck using a C++11 compatible compiler so features from C++14 and beyond are not available)