| L(s) = 1 | + (3.97 + 10.5i)2-s + (−50.6 + 20.9i)3-s + (−96.3 + 84.2i)4-s + (188. − 455. i)5-s + (−423. − 452. i)6-s + (−120. − 120. i)7-s + (−1.27e3 − 684. i)8-s + (578. − 578. i)9-s + (5.57e3 + 185. i)10-s + (6.47e3 + 2.68e3i)11-s + (3.11e3 − 6.28e3i)12-s + (−1.76e3 − 4.25e3i)13-s + (798. − 1.76e3i)14-s + 2.70e4i·15-s + (2.17e3 − 1.62e4i)16-s − 2.33e4i·17-s + ⋯ |
| L(s) = 1 | + (0.351 + 0.936i)2-s + (−1.08 + 0.448i)3-s + (−0.752 + 0.658i)4-s + (0.674 − 1.62i)5-s + (−0.800 − 0.856i)6-s + (−0.133 − 0.133i)7-s + (−0.881 − 0.472i)8-s + (0.264 − 0.264i)9-s + (1.76 + 0.0587i)10-s + (1.46 + 0.607i)11-s + (0.519 − 1.05i)12-s + (−0.222 − 0.537i)13-s + (0.0778 − 0.171i)14-s + 2.06i·15-s + (0.132 − 0.991i)16-s − 1.15i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.14631 - 0.210357i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14631 - 0.210357i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.97 - 10.5i)T \) |
| good | 3 | \( 1 + (50.6 - 20.9i)T + (1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (-188. + 455. i)T + (-5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (120. + 120. i)T + 8.23e5iT^{2} \) |
| 11 | \( 1 + (-6.47e3 - 2.68e3i)T + (1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (1.76e3 + 4.25e3i)T + (-4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 + 2.33e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (7.82e3 + 1.89e4i)T + (-6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-4.43e4 + 4.43e4i)T - 3.40e9iT^{2} \) |
| 29 | \( 1 + (-5.25e4 + 2.17e4i)T + (1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 1.19e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (1.89e5 - 4.57e5i)T + (-6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (-3.75e5 + 3.75e5i)T - 1.94e11iT^{2} \) |
| 43 | \( 1 + (5.51e5 + 2.28e5i)T + (1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 1.23e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (7.94e5 + 3.29e5i)T + (8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (6.00e5 - 1.44e6i)T + (-1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (9.91e4 - 4.10e4i)T + (2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (-1.13e6 + 4.72e5i)T + (4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (-1.48e6 - 1.48e6i)T + 9.09e12iT^{2} \) |
| 73 | \( 1 + (-3.00e6 + 3.00e6i)T - 1.10e13iT^{2} \) |
| 79 | \( 1 - 6.45e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-3.56e5 - 8.61e5i)T + (-1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (8.55e6 + 8.55e6i)T + 4.42e13iT^{2} \) |
| 97 | \( 1 - 6.51e6T + 8.07e13T^{2} \) |
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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
−15.53219792016321499332953486305, −13.96656988865016665486599679984, −12.75075361657532974372156513765, −11.84312357458254571269228572925, −9.735663351582900683081131130123, −8.738333036073186931418918100548, −6.67147089187370171437737247861, −5.28078191187930471306069569865, −4.55712770578632962255648272077, −0.59795429900781288600149343638,
1.60968314269268204848795981360, 3.49496976713414284155224868230, 5.88476328553210429241221641134, 6.60416852671335367693847233028, 9.381915895698439253741146053692, 10.78960475880435550994070774317, 11.39637801452298560253057533310, 12.59749210804691744585887039756, 14.10655091485623618872800066744, 14.75784008835690562973226883946
