| L(s) = 1 | + (−10.3 − 4.66i)2-s + (9.79 − 23.6i)3-s + (84.4 + 96.1i)4-s + (215. − 89.3i)5-s + (−211. + 198. i)6-s + (−1.10e3 + 1.10e3i)7-s + (−422. − 1.38e3i)8-s + (1.08e3 + 1.08e3i)9-s + (−2.63e3 − 85.3i)10-s + (334. + 807. i)11-s + (3.10e3 − 1.05e3i)12-s + (8.88e3 + 3.67e3i)13-s + (1.65e4 − 6.23e3i)14-s − 5.97e3i·15-s + (−2.11e3 + 1.62e4i)16-s − 2.80e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.911 − 0.412i)2-s + (0.209 − 0.505i)3-s + (0.659 + 0.751i)4-s + (0.771 − 0.319i)5-s + (−0.399 + 0.374i)6-s + (−1.21 + 1.21i)7-s + (−0.291 − 0.956i)8-s + (0.495 + 0.495i)9-s + (−0.834 − 0.0269i)10-s + (0.0757 + 0.182i)11-s + (0.518 − 0.176i)12-s + (1.12 + 0.464i)13-s + (1.61 − 0.607i)14-s − 0.456i·15-s + (−0.128 + 0.991i)16-s − 1.38i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0347i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.31159 + 0.0228078i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31159 + 0.0228078i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (10.3 + 4.66i)T \) |
| good | 3 | \( 1 + (-9.79 + 23.6i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (-215. + 89.3i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (1.10e3 - 1.10e3i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (-334. - 807. i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (-8.88e3 - 3.67e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 + 2.80e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (-5.09e4 - 2.10e4i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-3.22e4 - 3.22e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (-2.90e4 + 7.02e4i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 1.91e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-1.00e5 + 4.16e4i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (-5.31e5 - 5.31e5i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (1.87e4 + 4.53e4i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 8.39e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-3.65e4 - 8.82e4i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (8.92e5 - 3.69e5i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (4.23e5 - 1.02e6i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (-2.80e5 + 6.76e5i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (7.04e5 - 7.04e5i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (4.04e6 + 4.04e6i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 - 4.52e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (4.39e6 + 1.82e6i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (-1.02e6 + 1.02e6i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 - 6.28e6T + 8.07e13T^{2} \) |
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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
−15.86753674104102136442619816165, −13.61794933138502038013433932453, −12.74422152016732116462549945364, −11.55439841722405586658971404211, −9.683432496419398844954721888621, −9.157808224224501778591972019658, −7.43293699229972382629034143428, −5.94173794715942841976473479360, −2.90836170943356226881210046447, −1.42068953229584165716961225179,
0.944180497899825656847541404049, 3.46269771807217797770467678052, 6.05759420350930914208031876460, 7.14237305414318765938554009300, 9.033882568208135765730452240913, 10.05175446843550075754034262012, 10.74824141257136874048555198101, 13.04333992849805682485876157784, 14.22846167291596537412433064244, 15.62608846628522633256642129438
