| L(s) = 1 | + 239. i·5-s + 396.·7-s + 1.96e3i·11-s + 934.·13-s − 1.90e3i·17-s + 9.04e3·19-s + 2.77e3i·23-s − 4.18e4·25-s − 2.46e4i·29-s − 2.38e4·31-s + 9.51e4i·35-s − 3.72e4·37-s + 1.00e5i·41-s + 1.47e5·43-s + 1.02e4i·47-s + ⋯ |
| L(s) = 1 | + 1.91i·5-s + 1.15·7-s + 1.47i·11-s + 0.425·13-s − 0.386i·17-s + 1.31·19-s + 0.228i·23-s − 2.67·25-s − 1.01i·29-s − 0.799·31-s + 2.21i·35-s − 0.736·37-s + 1.45i·41-s + 1.85·43-s + 0.0984i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.133022837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.133022837\) |
| \(L(4)\) |
|
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 | \( 1 - 239. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 396.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.96e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 934.T + 4.82e6T^{2} \) |
| 17 | \( 1 + 1.90e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 9.04e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 2.77e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.46e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 2.38e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 3.72e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 1.00e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.47e5T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.02e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.45e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.90e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 1.23e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 4.73e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + 1.72e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 9.25e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + 6.25e3T + 2.43e11T^{2} \) |
| 83 | \( 1 + 3.19e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 7.40e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 3.48e5T + 8.32e11T^{2} \) |
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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
−11.98522679462231168492473529922, −11.27036265522171303279722539629, −10.41770615273118670981101695921, −9.462196667193427300169006910220, −7.64167554073040520355532659755, −7.27177209433232672222198210950, −5.89852485506347232875492388050, −4.40641052260461705621281202815, −2.95440215731255910004081040429, −1.75287984754935376072382838571,
0.68156501334412547042178403029, 1.56089266062726652196791432536, 3.75927993922860483454624713747, 5.06482372079677693801077492200, 5.69393413430766083567490545725, 7.72030314099271108686003088109, 8.595567558157806675649248820328, 9.127663872586411018987384344872, 10.78690280896606624464459729481, 11.70020965708003046742165724466
