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Many results in this direction can be found in the paper

B. A. Magurn: Negative (K)-theory of generalized quaternion groups and binary polyhedral groups, Commun. Algebra 41, No. 11, 4146-4160 (2013). ZBL1284.19004.

In particular, you can have a look at

Corollary (p.4155) Given the dicyclic group $Q_n$, the group $K_{-1}(\mathbb{Z}Q_n)$ is torsion free if and only if $n$ is a power of a prime in $3 + 4 \mathbb{Z}$ or $n=2$.

All the other occurrences for $n$ provide an infinite series of examples with $s\neq 0$. The paper also contains explicit calculations for $s$.

Theorem 5 (p.4155) The group $K_{-1}$ for binary polyhedral groups is as follows: $$K_{-1}(\mathbb{Z} \tilde{\mathsf{A}}_4)= \mathbb{Z}, \quad K_{-1}(\mathbb{Z} \tilde{\mathsf{S}}_4)= \mathbb{Z}\oplus \mathbb{Z}_2, \quad K_{-1}(\mathbb{Z} \tilde{\mathsf{A}}_5)= \mathbb{Z}^2\oplus \mathbb{Z}_2$$

This yields two further examples with $s=1$.

Many results in this direction can be found in the paper

B. A. Magurn: Negative (K)-theory of generalized quaternion groups and binary polyhedral groups, Commun. Algebra 41, No. 11, 4146-4160 (2013). ZBL1284.19004.

In particular, you can have a look at

Corollary (p.4155) Given the dicyclic group $\mathsf{Q}_n$, the group $K_{-1}(\mathbb{Z} \mathsf{Q}_n)$ is torsion free if and only if $n$ is a power of a prime in $3 + 4 \mathbb{Z}$ or $n=2$.

All the other occurrences for $n$ provide an infinite series of examples with $s\neq 0$. The paper also contains explicit calculations for $s$.

Theorem 5 (p.4155) The group $K_{-1}$ for binary polyhedral groups is as follows: $$K_{-1}(\mathbb{Z} \tilde{\mathsf{A}}_4)= \mathbb{Z}, \quad K_{-1}(\mathbb{Z} \tilde{\mathsf{S}}_4)= \mathbb{Z}\oplus \mathbb{Z}_2, \quad K_{-1}(\mathbb{Z} \tilde{\mathsf{A}}_5)= \mathbb{Z}^2\oplus \mathbb{Z}_2$$

This yields two further examples with $s=1$.

Many results in this direction can be found in the paper

B. A. Magurn: Negative (K)-theory of generalized quaternion groups and binary polyhedral groups, Commun. Algebra 41, No. 11, 4146-4160 (2013). ZBL1284.19004.

In particular, you can have a look at

Corollary (p.4155) Given the dicyclic group $Q_n$, the group $K_{-1}(\mathbb{Z}Q_n)$ is torsion free if and only if $n$ is a power of a prime in $3 + 4 \mathbb{Z}$ or $n=2$.

All the other occurrences for $n$ provide an infinite series of examples with $s\neq 0$. The paper also contains explicit calculations for $s$.

Theorem (p.4155) The group $K_{-1}$ for binary polyhedral groups is as follows: $$K_{-1}(\mathbb{Z} \tilde{\mathsf{A}}_4)= \mathbb{Z}, \quad K_{-1}(\mathbb{Z} \tilde{\mathsf{S}}_4)= \mathbb{Z}\oplus \mathbb{Z}_2, \quad K_{-1}(\mathbb{Z} \tilde{\mathsf{A}}_5)= \mathbb{Z}^2\oplus \mathbb{Z}_2$$

This yields two further examples with $s=1$.

Many results in this direction can be found in the paper

B. A. Magurn: Negative (K)-theory of generalized quaternion groups and binary polyhedral groups, Commun. Algebra 41, No. 11, 4146-4160 (2013). ZBL1284.19004.

In particular, you can have a look at

Corollary (p.4155) Given the dicyclic group $Q_n$, the group $K_{-1}(\mathbb{Z}Q_n)$ is torsion free if and only if $n$ is a power of a prime in $3 + 4 \mathbb{Z}$ or $n=2$.

All the other occurrences for $n$ provide an infinite series of examples with $s\neq 0$. The paper also contains explicit calculations for $s$.

Theorem 5 (p.4155) The group $K_{-1}$ for binary polyhedral groups is as follows: $$K_{-1}(\mathbb{Z} \tilde{\mathsf{A}}_4)= \mathbb{Z}, \quad K_{-1}(\mathbb{Z} \tilde{\mathsf{S}}_4)= \mathbb{Z}\oplus \mathbb{Z}_2, \quad K_{-1}(\mathbb{Z} \tilde{\mathsf{A}}_5)= \mathbb{Z}^2\oplus \mathbb{Z}_2$$

This yields two further examples with $s=1$.

Many results in this direction can be found in the paper.

Magurn, Bruce A., Negative (K)-theory of generalized quaternion groups and binary polyhedral groups, Commun. Algebra 41, No. 11, 4146-4160 (2013). ZBL1284.19004.

Corollary (p.4155) Given the dicyclic group $Q_n$, the group $K_{-1}\mathbb{Z}(Q_n)$ is torsion free if and only if $n$ is a power of a prime in $3 + 4 \mathbb{Z}$ or $n=2$.

Theorem (p.4155) The group $K_{-1}$ for binary polyhedral groups is as follows: $$K_{-1}(\mathbb{Z} \tilde{\mathsf{A}}_4)= \mathbb{Z}, \quad K_{-1}(\mathbb{Z} \tilde{\mathsf{A}}_4)= \mathbb{Z}\oplus \mathbb{Z}_2, \quad K_{-1}(\mathbb{Z} \tilde{\mathsf{A}}_5)= \mathbb{Z}^2\oplus \mathbb{Z}_2$$

Many results in this direction can be found in the paper

B. A. Magurn: Negative (K)-theory of generalized quaternion groups and binary polyhedral groups, Commun. Algebra 41, No. 11, 4146-4160 (2013). ZBL1284.19004.

In particular, you can have a look at

Corollary (p.4155) Given the dicyclic group $Q_n$, the group $K_{-1}(\mathbb{Z}Q_n)$ is torsion free if and only if $n$ is a power of a prime in $3 + 4 \mathbb{Z}$ or $n=2$.

All the other occurrences for $n$ provide an infinite series of examples with $s\neq 0$. The paper also contains explicit calculations for $s$.

Theorem (p.4155) The group $K_{-1}$ for binary polyhedral groups is as follows: $$K_{-1}(\mathbb{Z} \tilde{\mathsf{A}}_4)= \mathbb{Z}, \quad K_{-1}(\mathbb{Z} \tilde{\mathsf{S}}_4)= \mathbb{Z}\oplus \mathbb{Z}_2, \quad K_{-1}(\mathbb{Z} \tilde{\mathsf{A}}_5)= \mathbb{Z}^2\oplus \mathbb{Z}_2$$

This yields two further examples with $s=1$.