| L(s) = 1 | + (0.362 − 11.3i)2-s + (−6.67 + 16.1i)3-s + (−127. − 8.20i)4-s + (−71.5 + 29.6i)5-s + (179. + 81.3i)6-s + (108. − 108. i)7-s + (−139. + 1.44e3i)8-s + (1.33e3 + 1.33e3i)9-s + (309. + 820. i)10-s + (1.53e3 + 3.70e3i)11-s + (984. − 2.00e3i)12-s + (2.09e3 + 868. i)13-s + (−1.19e3 − 1.27e3i)14-s − 1.35e3i·15-s + (1.62e4 + 2.09e3i)16-s + 492. i·17-s + ⋯ |
| L(s) = 1 | + (0.0320 − 0.999i)2-s + (−0.142 + 0.344i)3-s + (−0.997 − 0.0640i)4-s + (−0.256 + 0.106i)5-s + (0.339 + 0.153i)6-s + (0.120 − 0.120i)7-s + (−0.0960 + 0.995i)8-s + (0.608 + 0.608i)9-s + (0.0978 + 0.259i)10-s + (0.347 + 0.839i)11-s + (0.164 − 0.334i)12-s + (0.264 + 0.109i)13-s + (−0.116 − 0.123i)14-s − 0.103i·15-s + (0.991 + 0.127i)16-s + 0.0243i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.27335 + 0.277253i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27335 + 0.277253i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.362 + 11.3i)T \) |
| good | 3 | \( 1 + (6.67 - 16.1i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (71.5 - 29.6i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (-108. + 108. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (-1.53e3 - 3.70e3i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (-2.09e3 - 868. i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 - 492. iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (-2.59e4 - 1.07e4i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-1.09e4 - 1.09e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (6.65e4 - 1.60e5i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 5.75e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + (1.58e5 - 6.54e4i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (-3.71e4 - 3.71e4i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (-5.52e4 - 1.33e5i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 + 8.01e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-4.10e5 - 9.91e5i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (-1.80e6 + 7.48e5i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (-5.58e4 + 1.34e5i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (-5.44e5 + 1.31e6i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (-3.40e6 + 3.40e6i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (1.81e6 + 1.81e6i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 + 4.67e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-1.18e6 - 4.91e5i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (3.93e6 - 3.93e6i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 + 1.32e7T + 8.07e13T^{2} \) |
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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
−15.22589366777191242412119861537, −13.91722900108166601400333898101, −12.71058029477432480719784550083, −11.49734400866103831645923155393, −10.39180929237418941973891191619, −9.286310759963659296171443517299, −7.53235183243119684163839938788, −5.10031810334680526016368616722, −3.73118734813116256966431514303, −1.62682679270137081129861853589,
0.68725017529060988263625404219, 3.90943255540255416661359877712, 5.73801022080856537214576530225, 7.03530782067573535232588865657, 8.372663111283275449841192613589, 9.695227769875214674847969848253, 11.65720149911538318684539864697, 12.97393426713966519971564445027, 14.06102948404872684964904677937, 15.35936779872394158595938386166
