| L(s) = 1 | − 137. i·5-s − 808.·7-s + 8.24e3i·11-s + 3.40e3i·13-s − 2.07e4·17-s − 6.30e3i·19-s + 6.88e4·23-s + 5.93e4·25-s − 4.20e4i·29-s + 2.14e5·31-s + 1.10e5i·35-s − 1.82e5i·37-s − 3.95e5·41-s − 4.33e5i·43-s − 1.32e6·47-s + ⋯ |
| L(s) = 1 | − 0.490i·5-s − 0.890·7-s + 1.86i·11-s + 0.430i·13-s − 1.02·17-s − 0.210i·19-s + 1.18·23-s + 0.759·25-s − 0.320i·29-s + 1.29·31-s + 0.436i·35-s − 0.593i·37-s − 0.897·41-s − 0.831i·43-s − 1.86·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5732715095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5732715095\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 137. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 808.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 8.24e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 3.40e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + 2.07e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 6.30e3iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 6.88e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.20e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 2.14e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.82e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 3.95e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.33e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 1.32e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.25e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 2.06e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 3.11e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 1.22e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 1.63e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.43e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.60e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.38e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 6.88e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.48e6T + 8.07e13T^{2} \) |
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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
−10.04189122390128820990694516217, −9.415696624228304562642843460132, −8.508040117371719925056714246358, −7.08520589606881927183023710940, −6.59672872933488876052788577228, −5.01361876807021700694005777123, −4.29810841315780397964945016426, −2.81374272119591846324634569963, −1.63121717091632460758230181755, −0.14568345018208326685978524349,
0.990870750321399494764425554730, 2.86571943524292631902866159766, 3.36170912125926655155714021522, 4.94852899637524116339031455442, 6.23485221624755077306093730395, 6.73454183336027354909891474449, 8.189158957191076901498828235039, 8.929580589607107386686260998051, 10.06531493415959015116478951218, 10.93217375719902052192369886762
