| L(s) = 1 | + 2.58·2-s − 1.31·4-s − 5·5-s − 22.8·7-s − 24.0·8-s − 12.9·10-s − 11.0·11-s − 11.6·13-s − 59.1·14-s − 51.7·16-s − 10.0·17-s + 117.·19-s + 6.59·20-s − 28.6·22-s − 172.·23-s + 25·25-s − 30.0·26-s + 30.1·28-s − 178.·29-s + 140.·31-s + 59.0·32-s − 26.0·34-s + 114.·35-s + 250.·37-s + 304.·38-s + 120.·40-s − 361.·41-s + ⋯ |
| L(s) = 1 | + 0.913·2-s − 0.164·4-s − 0.447·5-s − 1.23·7-s − 1.06·8-s − 0.408·10-s − 0.303·11-s − 0.248·13-s − 1.12·14-s − 0.808·16-s − 0.143·17-s + 1.42·19-s + 0.0737·20-s − 0.277·22-s − 1.56·23-s + 0.200·25-s − 0.226·26-s + 0.203·28-s − 1.14·29-s + 0.814·31-s + 0.326·32-s − 0.131·34-s + 0.552·35-s + 1.11·37-s + 1.30·38-s + 0.476·40-s − 1.37·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 2 | \( 1 - 2.58T + 8T^{2} \) |
| 7 | \( 1 + 22.8T + 343T^{2} \) |
| 11 | \( 1 + 11.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 11.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 117.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 250.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 361.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 600.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 201.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 415.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 54.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 531.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 933.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 560.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 810.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 538.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 686.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 714.T + 9.12e5T^{2} \) |
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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
−12.37663174797230766104234839769, −11.67814637656873645324046210556, −10.06327599451344043598990676847, −9.259689462649478747742999760843, −7.84159135132084991879744901349, −6.47360098252878396449022845602, −5.39309285183651874910948327425, −4.01605913178410730470129241910, −2.99192579651549168603320142315, 0,
2.99192579651549168603320142315, 4.01605913178410730470129241910, 5.39309285183651874910948327425, 6.47360098252878396449022845602, 7.84159135132084991879744901349, 9.259689462649478747742999760843, 10.06327599451344043598990676847, 11.67814637656873645324046210556, 12.37663174797230766104234839769
