As Euler demonstrated it is possible to represent the result of the Riemann Zeta Function for all even values of s (2, 4, 6,...) in terms of an expression involving pi, i.e. (pi^s)/k where k is a rational number.
However - as is well known - a similar sort of expression cannot be provided for the corresponding odd values of s (3, 5, 7,....)
So, from my perspective the first task is to explain in qualitative terms why this situation arises!
From the psychological perspective each value of s represents a "higher" dimension of understanding (which traditionally has been associated with contemplative type development).
In this context a crucial distinction can be made as between the odd and even numbers which is very revealing.
Whereas the odd numbers represent a new level of (rational) differentiation, the even numbers represent, by contrast, new equilibrium states of (intuitive) integration.
Now once again, the conventional (1-dimensional) level is viewed as a means of (rational) differentiation of meaning. Though informally the importance of supporting intuition may be admitted, this is not formally included in interpretation. And as we have seen this leads to the inevitable reduction of qualitative to quantitative type analysis.
In general terms differentiation is based on the separation of poles in experience, whereas integration is based on their complementarity (and ultimate identity).
From another perspective there always remains a certain linear character to the differentiated aspect of understanding (even at higher levels) while the integral aspect is always characterised by complementarity.
As stated previously the qualitative structure of a dimension is inversely related to its corresponding root structure. So for example the two roots of unity are + 1 and - 1 respectively; in corresponding inverse qualitative fashion, 2-dimensional appreciation is characterised by the complementarity of opposite poles of form. So with the quantitative interpretation of roots we employ a (linear) either/or logic; with the corresponding qualitative interpretation of dimensions we employ a (circular) both/and logic.
Now when we look at the root structure for all even numbered roots, they are characterised by the complementarity of opposites where half of the roots can be balanced by the other half (that represents the negative of all roots in the first half).
Now the very nature of such complementary understanding is that it is perfectly circular in nature. However because customary mathematical understanding is based on linear type distinctions we can only attempt to convey the nature of such circular understanding (which ultimately is of a purely intuitive nature) in a reduced linear fashion.
So from a rational perspective the nature or 2-dimensional understanding entails the relationship as between what is circular and linear. In corresponding quantitative fashion the very nature of pi involves the relationship as between circle and line (i.e. as the ratio of circular circumference to line diameter).
Thus the qualitative nature of all even dimensions (as representing the refined rational linear manner of conveying circular meaning) is perfectly replicated in quantitative terms in the numerical expressions of the Zeta Function for even dimensional values that are positive.
When we consider the structure of odd number roots, perfect quantitative complementarity does not exist. This is evidenced by the fact that the odd numbered roots cannot be arranged in a fully complementary manner! In corresponding fashion, qualitative complementarity likewise does not exist. Therefore in both cases, they represent a situation of broken symmetry.
In fact with odd numbered roots, + 1 is always one root obtained that in a sense remains separated from all other roots where a complementary pattern does exist with respect to the imaginary parts of these roots. So the broken symmetry relates here to a distinction as between the behaviour of real and imaginary parts. Likewise in qualitative terms, at the higher odd numbered dimensions a degree of linear understanding is maintained thereby enabling consious differentiation of phenomena that is ultimately inconsistent with the (unconscious) holistic understanding of these dimensions. Thus the imbalance of conscious and unconscious remaining requires progression to the next (even) higher dimension where a new level of integration can be obtained.
So from a psychological perspective, again it is easy to suggest why such broken symmetry might exist.
In development each new dimensional stage of integration (where a temporary equilibrium is reached) follows a corresponding previous dimensional stage of differentiation (where such equilibrium is temporarily broken).
Therefore the (positive) odd-numbered dimensions represent a situation where a new configuration of both (refined) rational and intuitive understanding go hand in hand. In other words experience is literally somewhat uneven whereby the relationship as between both linear and circular type interpretation entails a degree of confusion. When such confusion is then unravelled one moves on to the next integral stage (represented by an even numbered dimension). And then the deepening of such integral awareness requires the further progression through ever "higher" alternating odd and even-numbered dimensions.
Now ultimately with very high numbered dimensions the differentiated rational element of understanding becomes so refined that it is no longer (explicitly) separable from the intuitive. So here both integral and differentiated understanding closely approximate (where the discrete becomes inseparable from the continuous).
Remarkably here we have the qualitative correspondent of e. As is well known both the differentiated and integrated expressions of e^x remain the same.
The same e also plays an enormous role with respect to the distribution of prime numbers. In corresponding qualitative fashion e represents the dynamic state where (integral) intuition cannot be distinguished from (differentiated) reason. This plays an enormous role with respect to the holistic mastery of primitive desires (so that their discrete involuntary nature can be controlled). And it is this same experience that characterises the qualitative appreciation of the holistic nature of prime numbers!