| L(s) = 1 | + 5-s + 2·7-s − 2·13-s − 2·17-s + 4·19-s + 23-s + 25-s − 8·31-s + 2·35-s + 4·37-s − 4·41-s − 6·43-s − 3·49-s − 2·53-s + 6·59-s − 6·61-s − 2·65-s − 10·67-s − 6·71-s − 14·73-s − 4·79-s − 4·83-s − 2·85-s + 14·89-s − 4·91-s + 4·95-s + 16·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s − 1.43·31-s + 0.338·35-s + 0.657·37-s − 0.624·41-s − 0.914·43-s − 3/7·49-s − 0.274·53-s + 0.781·59-s − 0.768·61-s − 0.248·65-s − 1.22·67-s − 0.712·71-s − 1.63·73-s − 0.450·79-s − 0.439·83-s − 0.216·85-s + 1.48·89-s − 0.419·91-s + 0.410·95-s + 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.29056915003097, −15.63089070607925, −14.89756023184515, −14.60208089749862, −14.09280029383273, −13.27838703787839, −13.10238564662568, −12.24518533139104, −11.60136138273429, −11.32899656691053, −10.45815026341877, −10.11357829716041, −9.244401724633446, −8.974579850327464, −8.137096296259930, −7.532075287592918, −7.056289873847685, −6.255803248232793, −5.571063020904813, −4.982284699556565, −4.471014428831327, −3.517675591201600, −2.790115366686079, −1.921968946648558, −1.306511668208382, 0,
1.306511668208382, 1.921968946648558, 2.790115366686079, 3.517675591201600, 4.471014428831327, 4.982284699556565, 5.571063020904813, 6.255803248232793, 7.056289873847685, 7.532075287592918, 8.137096296259930, 8.974579850327464, 9.244401724633446, 10.11357829716041, 10.45815026341877, 11.32899656691053, 11.60136138273429, 12.24518533139104, 13.10238564662568, 13.27838703787839, 14.09280029383273, 14.60208089749862, 14.89756023184515, 15.63089070607925, 16.29056915003097
