| L(s) = 1 | + 2-s + 1.01·3-s + 4-s + 3.08·5-s + 1.01·6-s + 1.61·7-s + 8-s − 1.96·9-s + 3.08·10-s − 1.92·11-s + 1.01·12-s − 5.39·13-s + 1.61·14-s + 3.13·15-s + 16-s + 6.81·17-s − 1.96·18-s − 19-s + 3.08·20-s + 1.64·21-s − 1.92·22-s + 2.24·23-s + 1.01·24-s + 4.52·25-s − 5.39·26-s − 5.04·27-s + 1.61·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.586·3-s + 0.5·4-s + 1.38·5-s + 0.414·6-s + 0.610·7-s + 0.353·8-s − 0.656·9-s + 0.976·10-s − 0.581·11-s + 0.293·12-s − 1.49·13-s + 0.431·14-s + 0.809·15-s + 0.250·16-s + 1.65·17-s − 0.464·18-s − 0.229·19-s + 0.690·20-s + 0.357·21-s − 0.410·22-s + 0.467·23-s + 0.207·24-s + 0.905·25-s − 1.05·26-s − 0.970·27-s + 0.305·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.528928412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.528928412\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 1.01T + 3T^{2} \) |
| 5 | \( 1 - 3.08T + 5T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 + 5.39T + 13T^{2} \) |
| 17 | \( 1 - 6.81T + 17T^{2} \) |
| 23 | \( 1 - 2.24T + 23T^{2} \) |
| 29 | \( 1 - 7.76T + 29T^{2} \) |
| 31 | \( 1 - 0.778T + 31T^{2} \) |
| 37 | \( 1 - 3.25T + 37T^{2} \) |
| 41 | \( 1 - 8.93T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 1.95T + 47T^{2} \) |
| 53 | \( 1 - 8.08T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 4.28T + 61T^{2} \) |
| 67 | \( 1 - 4.68T + 67T^{2} \) |
| 71 | \( 1 + 2.72T + 71T^{2} \) |
| 73 | \( 1 - 0.822T + 73T^{2} \) |
| 79 | \( 1 - 6.10T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 0.942T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.73174231962433179546338549621, −7.25488032381132375894255558030, −6.12531250624975189776908224073, −5.73014670451682645281904998227, −5.06083911010923043096962319526, −4.51462719789300287041725028143, −3.25400531141056940492720809973, −2.59012284607199419273169006975, −2.21309007011915453579468368472, −1.04551105144370405978280025518,
1.04551105144370405978280025518, 2.21309007011915453579468368472, 2.59012284607199419273169006975, 3.25400531141056940492720809973, 4.51462719789300287041725028143, 5.06083911010923043096962319526, 5.73014670451682645281904998227, 6.12531250624975189776908224073, 7.25488032381132375894255558030, 7.73174231962433179546338549621
