| L(s) = 1 | + (4 + 6.92i)2-s + (1.40 − 2.42i)3-s + (−31.9 + 55.4i)4-s + (219. + 379. i)5-s + 22.4·6-s + (−893. + 159. i)7-s − 511.·8-s + (1.08e3 + 1.88e3i)9-s + (−1.75e3 + 3.03e3i)10-s + (2.74e3 − 4.74e3i)11-s + (89.7 + 155. i)12-s + 4.00e3·13-s + (−4.68e3 − 5.54e3i)14-s + 1.22e3·15-s + (−2.04e3 − 3.54e3i)16-s + (1.40e4 − 2.42e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (0.0299 − 0.0519i)3-s + (−0.249 + 0.433i)4-s + (0.784 + 1.35i)5-s + 0.0424·6-s + (−0.984 + 0.176i)7-s − 0.353·8-s + (0.498 + 0.862i)9-s + (−0.554 + 0.960i)10-s + (0.620 − 1.07i)11-s + (0.0149 + 0.0259i)12-s + 0.505·13-s + (−0.455 − 0.540i)14-s + 0.0940·15-s + (−0.125 − 0.216i)16-s + (0.691 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.21185 + 1.35828i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21185 + 1.35828i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 - 6.92i)T \) |
| 7 | \( 1 + (893. - 159. i)T \) |
| good | 3 | \( 1 + (-1.40 + 2.42i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-219. - 379. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (-2.74e3 + 4.74e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 - 4.00e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (-1.40e4 + 2.42e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.19e4 + 2.06e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-3.68e4 - 6.38e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + 9.87e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-2.37e4 + 4.11e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-5.00e4 - 8.66e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 - 4.89e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.99e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (4.81e5 + 8.33e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (9.18e5 - 1.59e6i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-7.25e3 + 1.25e4i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.01e6 + 1.75e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.48e6 + 2.57e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + 4.34e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (7.50e5 - 1.29e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-8.86e5 - 1.53e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + 1.57e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (4.39e6 + 7.61e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + 1.03e7T + 8.07e13T^{2} \) |
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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
−18.42634182811043816259025013637, −16.81772276001089276239757137124, −15.61791317685586967160050951619, −14.07689669394332432595389974952, −13.31112821586088938233604481527, −11.09465042917916327096418915311, −9.473131472118987447841868755781, −7.16450825419831758761776135565, −5.91424652926713435682847753543, −3.08070768577618156438021177871,
1.30395592989153819834418731726, 4.12352173667413607766903918345, 6.15846084625808745640319208809, 9.094562760099214155731921104719, 10.06045288837880495620269935330, 12.56821385544384384168193409621, 12.81147825846249849668843122245, 14.68822430910562637674368595932, 16.39933303402049304765745018468, 17.58210974974313546083170374489
