| L(s) = 1 | + 581.·5-s − 2.67e3i·7-s − 1.41e4i·11-s + 4.28e4·13-s + 1.40e5·17-s + 1.29e5i·19-s + 1.96e5i·23-s − 5.22e4·25-s + 3.55e5·29-s + 3.35e3i·31-s − 1.55e6i·35-s + 9.07e5·37-s + 3.06e6·41-s − 5.01e6i·43-s + 3.25e6i·47-s + ⋯ |
| L(s) = 1 | + 0.930·5-s − 1.11i·7-s − 0.966i·11-s + 1.49·13-s + 1.68·17-s + 0.995i·19-s + 0.703i·23-s − 0.133·25-s + 0.503·29-s + 0.00363i·31-s − 1.03i·35-s + 0.484·37-s + 1.08·41-s − 1.46i·43-s + 0.666i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.346529202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.346529202\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 581.T + 3.90e5T^{2} \) |
| 7 | \( 1 + 2.67e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 1.41e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 4.28e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.40e5T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.29e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.96e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 3.55e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 3.35e3iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 9.07e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.06e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 5.01e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 3.25e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 2.34e5T + 6.22e13T^{2} \) |
| 59 | \( 1 - 7.78e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 2.42e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 6.87e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 5.76e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 1.19e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 3.55e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 2.03e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 1.19e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 3.19e7T + 7.83e15T^{2} \) |
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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
−10.34430501596434530012829711209, −9.485436941836262475677705748292, −8.330176240888625341134838391953, −7.46778808849898393514950491595, −6.07724991195879795084092903031, −5.66805944187657280215863405325, −3.99228496240474911606966672119, −3.20613480544950991256855186236, −1.49448128609019521772906587136, −0.845843258059997347110128009766,
1.04840096545093274352028884994, 2.09614382620415974845065764374, 3.11961397685040269911936223127, 4.64660382932874412077010562309, 5.74956779645732532705955288065, 6.32649981595669051362449166700, 7.71557140300086290069798481474, 8.801583183571362623558654209748, 9.531138864687481804790815621373, 10.38429471469313833083517957912
