| 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
−11.45345164297682424455886661113, −10.70742964436294593492529366686, −9.708058815749463589789906776056, −8.860421851066845083584470683685, −7.51365717844054240919944881399, −7.02654015893774118133847841715, −5.65787377987782346227223430278, −4.79350027098646433848817811852, −3.62348394304690310877534641208, −2.49679039755020517234846591576,
0.858615631274671755417776633188, 2.27495594540916611182646647733, 3.66528259809704632627315604648, 4.24127843225993736935982755464, 5.80290730770847044977358435145, 7.41788341331332605848207347156, 7.81135352858366459137675823756, 8.798081184921359192640789221117, 10.06366719750600228099472739626, 11.02088662251173566723435452262
