| L(s) = 1 | + (0.743 + 0.103i)2-s + (−2.33 + 0.844i)3-s + (−1.38 − 0.393i)4-s + (−0.407 + 0.426i)5-s + (−1.82 + 0.385i)6-s + (3.30 − 1.02i)7-s + (−2.35 − 1.04i)8-s + (2.44 − 2.03i)9-s + (−0.347 + 0.274i)10-s + (−0.359 − 3.10i)11-s + (3.56 − 0.247i)12-s + (−1.70 − 4.40i)13-s + (2.56 − 0.417i)14-s + (0.592 − 1.34i)15-s + (0.796 + 0.492i)16-s + (1.50 − 3.84i)17-s + ⋯ |
| L(s) = 1 | + (0.525 + 0.0733i)2-s + (−1.34 + 0.487i)3-s + (−0.690 − 0.196i)4-s + (−0.182 + 0.190i)5-s + (−0.745 + 0.157i)6-s + (1.24 − 0.386i)7-s + (−0.834 − 0.368i)8-s + (0.815 − 0.677i)9-s + (−0.109 + 0.0869i)10-s + (−0.108 − 0.934i)11-s + (1.02 − 0.0713i)12-s + (−0.473 − 1.22i)13-s + (0.684 − 0.111i)14-s + (0.152 − 0.346i)15-s + (0.199 + 0.123i)16-s + (0.363 − 0.931i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.541186 - 0.443334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.541186 - 0.443334i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-1.50 + 3.84i)T \) |
| good | 2 | \( 1 + (-0.743 - 0.103i)T + (1.92 + 0.547i)T^{2} \) |
| 3 | \( 1 + (2.33 - 0.844i)T + (2.30 - 1.91i)T^{2} \) |
| 5 | \( 1 + (0.407 - 0.426i)T + (-0.230 - 4.99i)T^{2} \) |
| 7 | \( 1 + (-3.30 + 1.02i)T + (5.77 - 3.95i)T^{2} \) |
| 11 | \( 1 + (0.359 + 3.10i)T + (-10.7 + 2.51i)T^{2} \) |
| 13 | \( 1 + (1.70 + 4.40i)T + (-9.60 + 8.75i)T^{2} \) |
| 19 | \( 1 + (-0.109 - 0.784i)T + (-18.2 + 5.19i)T^{2} \) |
| 23 | \( 1 + (1.77 + 5.74i)T + (-18.9 + 12.9i)T^{2} \) |
| 29 | \( 1 + (1.18 + 0.938i)T + (6.63 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-1.05 - 0.0244i)T + (30.9 + 1.43i)T^{2} \) |
| 37 | \( 1 + (7.12 - 6.19i)T + (5.11 - 36.6i)T^{2} \) |
| 41 | \( 1 + (8.75 + 4.11i)T + (26.1 + 31.5i)T^{2} \) |
| 43 | \( 1 + (1.78 + 0.419i)T + (38.4 + 19.1i)T^{2} \) |
| 47 | \( 1 + (-8.94 + 0.828i)T + (46.1 - 8.63i)T^{2} \) |
| 53 | \( 1 + (-1.69 - 2.03i)T + (-9.73 + 52.0i)T^{2} \) |
| 59 | \( 1 + (3.59 + 2.45i)T + (21.3 + 55.0i)T^{2} \) |
| 61 | \( 1 + (2.12 + 10.0i)T + (-55.8 + 24.6i)T^{2} \) |
| 67 | \( 1 + (-0.659 + 0.873i)T + (-18.3 - 64.4i)T^{2} \) |
| 71 | \( 1 + (-2.84 + 5.38i)T + (-40.1 - 58.5i)T^{2} \) |
| 73 | \( 1 + (5.79 - 4.16i)T + (23.1 - 69.2i)T^{2} \) |
| 79 | \( 1 + (-2.98 + 11.5i)T + (-69.0 - 38.4i)T^{2} \) |
| 83 | \( 1 + (-10.1 + 0.470i)T + (82.6 - 7.65i)T^{2} \) |
| 89 | \( 1 + (-4.90 + 12.6i)T + (-65.7 - 59.9i)T^{2} \) |
| 97 | \( 1 + (-14.1 - 7.44i)T + (54.8 + 80.0i)T^{2} \) |
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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
−11.65124462511113768995383762287, −10.66446059188623669779902772434, −10.17797106176585911971209477144, −8.725266737249245874683224385760, −7.67833934460264847091816179056, −6.21585957186780883073306872456, −5.16244494693833128460279735169, −4.89409603218724128561143097098, −3.46312149328641889575172086115, −0.55160224583897467279964426545,
1.78276309745005964379668337845, 4.15813266859898594927386320720, 4.97532313162469440904963998058, 5.70172474399679530910292779875, 6.99131698642466439394795216359, 8.071617678804483722691400437472, 9.129714404646081518187410957710, 10.37873489848991903810666272935, 11.58921343769001406405061763302, 12.00875727646678470567357073352
