In C++ you cannot combine declaration and initialization. When you do not specify constructor parameters of operator_list, you do not call its default constructor: you simply declare it. You need to also initialize it in the corresponding C++ file, like this:
Equation.h
class Equation {
    static Dictionary operator_list;
};
Equation.cpp:
Dictionary Equation::operator_list(1);
Note the absence of static in the CPP file: it is not there by design. The compiler already knows from the declaration that operator_list is static.
Edit: You have a choice with static constant members of integral and enumerated types: you can initialize them in the CPP file as in the example above, or you can give them a value in the header. You still need to define that member in your C++ file, but you must not give it a value at the definition time.