| L(s) = 1 | + (−10.6 − 11.9i)2-s + (−20.6 − 49.8i)3-s + (−28.0 + 254. i)4-s + (400. − 966. i)5-s + (−373. + 778. i)6-s + (679. − 679. i)7-s + (3.33e3 − 2.38e3i)8-s + (2.57e3 − 2.57e3i)9-s + (−1.57e4 + 5.54e3i)10-s + (8.34e3 − 2.01e4i)11-s + (1.32e4 − 3.85e3i)12-s + (1.29e3 + 3.13e3i)13-s + (−1.53e4 − 844. i)14-s − 5.64e4·15-s + (−6.39e4 − 1.42e4i)16-s + 1.11e5i·17-s + ⋯ |
| L(s) = 1 | + (−0.667 − 0.744i)2-s + (−0.255 − 0.615i)3-s + (−0.109 + 0.993i)4-s + (0.640 − 1.54i)5-s + (−0.288 + 0.600i)6-s + (0.282 − 0.282i)7-s + (0.813 − 0.581i)8-s + (0.392 − 0.392i)9-s + (−1.57 + 0.554i)10-s + (0.570 − 1.37i)11-s + (0.640 − 0.186i)12-s + (0.0454 + 0.109i)13-s + (−0.399 − 0.0219i)14-s − 1.11·15-s + (−0.975 − 0.218i)16-s + 1.34i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0783i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.996 - 0.0783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0491289 + 1.25183i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0491289 + 1.25183i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (10.6 + 11.9i)T \) |
| good | 3 | \( 1 + (20.6 + 49.8i)T + (-4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (-400. + 966. i)T + (-2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (-679. + 679. i)T - 5.76e6iT^{2} \) |
| 11 | \( 1 + (-8.34e3 + 2.01e4i)T + (-1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (-1.29e3 - 3.13e3i)T + (-5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 - 1.11e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (8.89e4 - 3.68e4i)T + (1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (9.77e4 + 9.77e4i)T + 7.83e10iT^{2} \) |
| 29 | \( 1 + (-7.83e5 + 3.24e5i)T + (3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 - 1.30e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (-7.97e5 + 1.92e6i)T + (-2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (-5.39e5 + 5.39e5i)T - 7.98e12iT^{2} \) |
| 43 | \( 1 + (-2.98e5 + 7.20e5i)T + (-8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 + 3.01e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + (-1.02e7 - 4.22e6i)T + (4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (1.51e7 + 6.28e6i)T + (1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (1.35e7 - 5.60e6i)T + (1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (-1.15e7 - 2.78e7i)T + (-2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (1.42e7 - 1.42e7i)T - 6.45e14iT^{2} \) |
| 73 | \( 1 + (-3.46e7 + 3.46e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 + 1.31e6T + 1.51e15T^{2} \) |
| 83 | \( 1 + (-1.49e7 + 6.19e6i)T + (1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (-2.83e7 - 2.83e7i)T + 3.93e15iT^{2} \) |
| 97 | \( 1 - 1.05e8T + 7.83e15T^{2} \) |
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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
−13.84045745885538338582810782936, −12.79436549465303534985861783069, −12.09102401946255470242884389316, −10.55080512871800828422852811930, −9.055121201486731575031392550366, −8.218843206341736940954308641801, −6.19462094039397566248363200226, −4.13702713127702628244897295953, −1.58174179202029748793327981048, −0.73821938399461262953251469664,
2.13249380239804085846145363874, 4.79264867709397847815656476899, 6.43675403729097481545933473910, 7.47064072391036457884810238060, 9.551520181044659984537481594297, 10.22104823661485035559667731383, 11.37638167506220587750152267321, 13.72107709740612667395710672305, 14.84451785514231799718434966282, 15.45297996104547675442305596512
