| L(s) = 1 | − 5-s − 4·7-s − 4·11-s − 2·13-s + 2·17-s + 19-s − 8·23-s + 25-s − 6·29-s − 4·31-s + 4·35-s − 10·37-s + 2·41-s − 12·43-s + 9·49-s − 6·53-s + 4·55-s − 10·61-s + 2·65-s + 4·67-s − 8·71-s + 2·73-s + 16·77-s + 12·79-s − 8·83-s − 2·85-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.229·19-s − 1.66·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.676·35-s − 1.64·37-s + 0.312·41-s − 1.82·43-s + 9/7·49-s − 0.824·53-s + 0.539·55-s − 1.28·61-s + 0.248·65-s + 0.488·67-s − 0.949·71-s + 0.234·73-s + 1.82·77-s + 1.35·79-s − 0.878·83-s − 0.216·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−16.55713211441445, −16.17267708266861, −15.68341254846806, −15.25488444581209, −14.54298021279244, −13.82843585375863, −13.35131683006762, −12.70441761959627, −12.36741547947524, −11.79058992102475, −11.01624018053720, −10.22218855354513, −10.04246268216197, −9.364011634391960, −8.680113275279937, −7.838077997115796, −7.514720864501513, −6.774431200154739, −6.098283514470747, −5.450962035925957, −4.835370725334566, −3.741782461821015, −3.406050080543009, −2.611529424630720, −1.730904613668053, 0, 0,
1.730904613668053, 2.611529424630720, 3.406050080543009, 3.741782461821015, 4.835370725334566, 5.450962035925957, 6.098283514470747, 6.774431200154739, 7.514720864501513, 7.838077997115796, 8.680113275279937, 9.364011634391960, 10.04246268216197, 10.22218855354513, 11.01624018053720, 11.79058992102475, 12.36741547947524, 12.70441761959627, 13.35131683006762, 13.82843585375863, 14.54298021279244, 15.25488444581209, 15.68341254846806, 16.17267708266861, 16.55713211441445
