| L(s) = 1 | + (−1.17 − 0.164i)2-s + (−0.867 + 0.313i)3-s + (−0.563 − 0.160i)4-s + (1.82 − 1.91i)5-s + (1.07 − 0.226i)6-s + (−0.135 + 0.0420i)7-s + (2.81 + 1.24i)8-s + (−1.65 + 1.37i)9-s + (−2.46 + 1.95i)10-s + (−0.132 − 1.14i)11-s + (0.539 − 0.0374i)12-s + (−1.23 − 3.19i)13-s + (0.166 − 0.0272i)14-s + (−0.984 + 2.23i)15-s + (−2.11 − 1.30i)16-s + (−2.87 + 2.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.832 − 0.116i)2-s + (−0.500 + 0.180i)3-s + (−0.281 − 0.0802i)4-s + (0.816 − 0.855i)5-s + (0.438 − 0.0924i)6-s + (−0.0513 + 0.0159i)7-s + (0.994 + 0.439i)8-s + (−0.551 + 0.457i)9-s + (−0.779 + 0.617i)10-s + (−0.0399 − 0.343i)11-s + (0.155 − 0.0108i)12-s + (−0.343 − 0.886i)13-s + (0.0446 − 0.00728i)14-s + (−0.254 + 0.575i)15-s + (−0.527 − 0.326i)16-s + (−0.698 + 0.715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.208542 - 0.397027i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.208542 - 0.397027i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (2.87 - 2.95i)T \) |
| good | 2 | \( 1 + (1.17 + 0.164i)T + (1.92 + 0.547i)T^{2} \) |
| 3 | \( 1 + (0.867 - 0.313i)T + (2.30 - 1.91i)T^{2} \) |
| 5 | \( 1 + (-1.82 + 1.91i)T + (-0.230 - 4.99i)T^{2} \) |
| 7 | \( 1 + (0.135 - 0.0420i)T + (5.77 - 3.95i)T^{2} \) |
| 11 | \( 1 + (0.132 + 1.14i)T + (-10.7 + 2.51i)T^{2} \) |
| 13 | \( 1 + (1.23 + 3.19i)T + (-9.60 + 8.75i)T^{2} \) |
| 19 | \( 1 + (1.04 + 7.51i)T + (-18.2 + 5.19i)T^{2} \) |
| 23 | \( 1 + (2.69 + 8.68i)T + (-18.9 + 12.9i)T^{2} \) |
| 29 | \( 1 + (0.879 + 0.696i)T + (6.63 + 28.2i)T^{2} \) |
| 31 | \( 1 + (0.122 + 0.00282i)T + (30.9 + 1.43i)T^{2} \) |
| 37 | \( 1 + (1.29 - 1.12i)T + (5.11 - 36.6i)T^{2} \) |
| 41 | \( 1 + (8.43 + 3.96i)T + (26.1 + 31.5i)T^{2} \) |
| 43 | \( 1 + (-11.2 - 2.63i)T + (38.4 + 19.1i)T^{2} \) |
| 47 | \( 1 + (8.34 - 0.772i)T + (46.1 - 8.63i)T^{2} \) |
| 53 | \( 1 + (-1.51 - 1.82i)T + (-9.73 + 52.0i)T^{2} \) |
| 59 | \( 1 + (-5.25 - 3.59i)T + (21.3 + 55.0i)T^{2} \) |
| 61 | \( 1 + (-0.796 - 3.77i)T + (-55.8 + 24.6i)T^{2} \) |
| 67 | \( 1 + (-4.53 + 6.00i)T + (-18.3 - 64.4i)T^{2} \) |
| 71 | \( 1 + (-5.59 + 10.6i)T + (-40.1 - 58.5i)T^{2} \) |
| 73 | \( 1 + (-12.4 + 8.94i)T + (23.1 - 69.2i)T^{2} \) |
| 79 | \( 1 + (1.20 - 4.62i)T + (-69.0 - 38.4i)T^{2} \) |
| 83 | \( 1 + (-7.89 + 0.365i)T + (82.6 - 7.65i)T^{2} \) |
| 89 | \( 1 + (4.12 - 10.6i)T + (-65.7 - 59.9i)T^{2} \) |
| 97 | \( 1 + (-11.6 - 6.16i)T + (54.8 + 80.0i)T^{2} \) |
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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
−11.00355384837797035368608865037, −10.57251617150189190345119911570, −9.521122622550758719559111931799, −8.714268103722122621137374283441, −8.085111306529205182554749080523, −6.41837574451483899635408093948, −5.27789271555177205135926154905, −4.62474630519683597671782497416, −2.30801762760783381012219291390, −0.46271940613486436954573095113,
1.87787437233083908052983070920, 3.71686681430774107920386329284, 5.30267356122436112040827731394, 6.43536163485742802966033567159, 7.21320403086825809324122331397, 8.394791428886761667129579262280, 9.592509596992312999930136323759, 9.895694450866537607399118320464, 11.08158973032547371009561261951, 11.90891022875438547436091811566
