| L(s) = 1 | − 2-s + 2.73·3-s + 4-s + 5-s − 2.73·6-s − 2.46·7-s − 8-s + 4.46·9-s − 10-s − 0.732·11-s + 2.73·12-s + 4.46·13-s + 2.46·14-s + 2.73·15-s + 16-s + 6.73·17-s − 4.46·18-s + 0.464·19-s + 20-s − 6.73·21-s + 0.732·22-s − 2.19·23-s − 2.73·24-s + 25-s − 4.46·26-s + 3.99·27-s − 2.46·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.57·3-s + 0.5·4-s + 0.447·5-s − 1.11·6-s − 0.931·7-s − 0.353·8-s + 1.48·9-s − 0.316·10-s − 0.220·11-s + 0.788·12-s + 1.23·13-s + 0.658·14-s + 0.705·15-s + 0.250·16-s + 1.63·17-s − 1.05·18-s + 0.106·19-s + 0.223·20-s − 1.46·21-s + 0.156·22-s − 0.457·23-s − 0.557·24-s + 0.200·25-s − 0.875·26-s + 0.769·27-s − 0.465·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.735040437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.735040437\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 + 0.732T + 11T^{2} \) |
| 13 | \( 1 - 4.46T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 - 0.464T + 19T^{2} \) |
| 23 | \( 1 + 2.19T + 23T^{2} \) |
| 29 | \( 1 - 6.26T + 29T^{2} \) |
| 31 | \( 1 + 9.73T + 31T^{2} \) |
| 37 | \( 1 + 6.19T + 37T^{2} \) |
| 41 | \( 1 + 4.46T + 41T^{2} \) |
| 47 | \( 1 - 5.26T + 47T^{2} \) |
| 53 | \( 1 + 5.46T + 53T^{2} \) |
| 59 | \( 1 + 6.53T + 59T^{2} \) |
| 61 | \( 1 + 1.73T + 61T^{2} \) |
| 67 | \( 1 - 5.19T + 67T^{2} \) |
| 71 | \( 1 + 3.26T + 71T^{2} \) |
| 73 | \( 1 - 9.19T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 - 9.12T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−10.72124447463230131315876686667, −9.943153749264273243336424181147, −9.309333234948764211906413474441, −8.526937701346771423689436457864, −7.79096932168984900649403720279, −6.76339813098567926288706145223, −5.65859646574230987807103901971, −3.64794824074334612399050328029, −3.00310925012367527400233058258, −1.59669078751643417199829952797,
1.59669078751643417199829952797, 3.00310925012367527400233058258, 3.64794824074334612399050328029, 5.65859646574230987807103901971, 6.76339813098567926288706145223, 7.79096932168984900649403720279, 8.526937701346771423689436457864, 9.309333234948764211906413474441, 9.943153749264273243336424181147, 10.72124447463230131315876686667
