| L(s) = 1 | + 2-s + 3.16·3-s + 4-s + 1.28·5-s + 3.16·6-s + 4.43·7-s + 8-s + 7.01·9-s + 1.28·10-s − 5.51·11-s + 3.16·12-s − 5.24·13-s + 4.43·14-s + 4.06·15-s + 16-s + 3.13·17-s + 7.01·18-s + 6.81·19-s + 1.28·20-s + 14.0·21-s − 5.51·22-s + 7.51·23-s + 3.16·24-s − 3.35·25-s − 5.24·26-s + 12.7·27-s + 4.43·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.82·3-s + 0.5·4-s + 0.574·5-s + 1.29·6-s + 1.67·7-s + 0.353·8-s + 2.33·9-s + 0.406·10-s − 1.66·11-s + 0.913·12-s − 1.45·13-s + 1.18·14-s + 1.04·15-s + 0.250·16-s + 0.760·17-s + 1.65·18-s + 1.56·19-s + 0.287·20-s + 3.05·21-s − 1.17·22-s + 1.56·23-s + 0.646·24-s − 0.670·25-s − 1.02·26-s + 2.44·27-s + 0.837·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.112120325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.112120325\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 3.16T + 3T^{2} \) |
| 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 - 4.43T + 7T^{2} \) |
| 11 | \( 1 + 5.51T + 11T^{2} \) |
| 13 | \( 1 + 5.24T + 13T^{2} \) |
| 17 | \( 1 - 3.13T + 17T^{2} \) |
| 19 | \( 1 - 6.81T + 19T^{2} \) |
| 23 | \( 1 - 7.51T + 23T^{2} \) |
| 29 | \( 1 + 5.25T + 29T^{2} \) |
| 37 | \( 1 + 6.48T + 37T^{2} \) |
| 41 | \( 1 + 6.98T + 41T^{2} \) |
| 43 | \( 1 + 5.80T + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 - 2.14T + 53T^{2} \) |
| 59 | \( 1 + 3.67T + 59T^{2} \) |
| 61 | \( 1 - 3.01T + 61T^{2} \) |
| 67 | \( 1 - 4.15T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 6.15T + 79T^{2} \) |
| 83 | \( 1 - 0.0222T + 83T^{2} \) |
| 89 | \( 1 - 6.89T + 89T^{2} \) |
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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
−7.939846425947520948345569320402, −7.39212958350100632984848269727, −7.22257982546900584799700206809, −5.42061032584166686454270984535, −5.17371604127327921598414264041, −4.55666263391026985542809830773, −3.34405158146650920740721582212, −2.84991072740965574120574030900, −2.07729271865779134715603529874, −1.48109765682191465909253112579,
1.48109765682191465909253112579, 2.07729271865779134715603529874, 2.84991072740965574120574030900, 3.34405158146650920740721582212, 4.55666263391026985542809830773, 5.17371604127327921598414264041, 5.42061032584166686454270984535, 7.22257982546900584799700206809, 7.39212958350100632984848269727, 7.939846425947520948345569320402
