| L(s) = 1 | + (1.17 − 1.62i)2-s + (1.88 − 2.33i)3-s + (−0.00519 − 0.0159i)4-s + (−3.22 − 4.43i)5-s + (−1.55 − 5.80i)6-s + (−1.72 − 5.31i)7-s + (7.59 + 2.46i)8-s + (−1.86 − 8.80i)9-s − 10.9·10-s + (−0.0470 − 0.0180i)12-s + (−7.86 − 5.71i)13-s + (−10.6 − 3.46i)14-s + (−16.4 − 0.863i)15-s + (12.9 − 9.44i)16-s + (10.5 + 14.4i)17-s + (−16.4 − 7.34i)18-s + ⋯ |
| L(s) = 1 | + (0.589 − 0.810i)2-s + (0.629 − 0.777i)3-s + (−0.00129 − 0.00399i)4-s + (−0.644 − 0.887i)5-s + (−0.259 − 0.967i)6-s + (−0.246 − 0.759i)7-s + (0.949 + 0.308i)8-s + (−0.207 − 0.978i)9-s − 1.09·10-s + (−0.00392 − 0.00150i)12-s + (−0.604 − 0.439i)13-s + (−0.761 − 0.247i)14-s + (−1.09 − 0.0575i)15-s + (0.812 − 0.590i)16-s + (0.617 + 0.850i)17-s + (−0.915 − 0.407i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.423934 - 2.37553i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.423934 - 2.37553i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.88 + 2.33i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-1.17 + 1.62i)T + (-1.23 - 3.80i)T^{2} \) |
| 5 | \( 1 + (3.22 + 4.43i)T + (-7.72 + 23.7i)T^{2} \) |
| 7 | \( 1 + (1.72 + 5.31i)T + (-39.6 + 28.8i)T^{2} \) |
| 13 | \( 1 + (7.86 + 5.71i)T + (52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (-10.5 - 14.4i)T + (-89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (5.76 - 17.7i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + 12.3iT - 529T^{2} \) |
| 29 | \( 1 + (-2.35 + 0.765i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (39.8 + 28.9i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (12.1 + 37.4i)T + (-1.10e3 + 804. i)T^{2} \) |
| 41 | \( 1 + (-53.7 - 17.4i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 - 43.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-54.9 - 17.8i)T + (1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (25.3 - 34.8i)T + (-868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (-85.6 + 27.8i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (24.9 - 18.1i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 - 34.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-22.0 - 30.4i)T + (-1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-3.74 - 11.5i)T + (-4.31e3 + 3.13e3i)T^{2} \) |
| 79 | \( 1 + (51.0 + 37.1i)T + (1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (5.70 + 7.84i)T + (-2.12e3 + 6.55e3i)T^{2} \) |
| 89 | \( 1 + 34.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-30.5 - 22.2i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−11.02088662251173566723435452262, −10.06366719750600228099472739626, −8.798081184921359192640789221117, −7.81135352858366459137675823756, −7.41788341331332605848207347156, −5.80290730770847044977358435145, −4.24127843225993736935982755464, −3.66528259809704632627315604648, −2.27495594540916611182646647733, −0.858615631274671755417776633188,
2.49679039755020517234846591576, 3.62348394304690310877534641208, 4.79350027098646433848817811852, 5.65787377987782346227223430278, 7.02654015893774118133847841715, 7.51365717844054240919944881399, 8.860421851066845083584470683685, 9.708058815749463589789906776056, 10.70742964436294593492529366686, 11.45345164297682424455886661113
