| L(s) = 1 | + (−2.24 − 1.07i)2-s + (1.53 + 0.811i)3-s + (2.61 + 3.27i)4-s + (0.419 − 0.389i)5-s + (−2.55 − 3.47i)6-s + (1.54 + 2.14i)7-s + (−1.21 − 5.31i)8-s + (1.68 + 2.48i)9-s + (−1.36 + 0.419i)10-s + (3.73 − 2.54i)11-s + (1.34 + 7.13i)12-s + (−4.93 + 3.36i)13-s + (−1.14 − 6.48i)14-s + (0.957 − 0.255i)15-s + (−1.15 + 5.04i)16-s + (1.91 + 0.288i)17-s + ⋯ |
| L(s) = 1 | + (−1.58 − 0.763i)2-s + (0.883 + 0.468i)3-s + (1.30 + 1.63i)4-s + (0.187 − 0.174i)5-s + (−1.04 − 1.41i)6-s + (0.584 + 0.811i)7-s + (−0.428 − 1.87i)8-s + (0.561 + 0.827i)9-s + (−0.430 + 0.132i)10-s + (1.12 − 0.768i)11-s + (0.387 + 2.05i)12-s + (−1.36 + 0.933i)13-s + (−0.306 − 1.73i)14-s + (0.247 − 0.0659i)15-s + (−0.287 + 1.26i)16-s + (0.463 + 0.0698i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.976253 + 0.141325i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.976253 + 0.141325i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.53 - 0.811i)T \) |
| 7 | \( 1 + (-1.54 - 2.14i)T \) |
| good | 2 | \( 1 + (2.24 + 1.07i)T + (1.24 + 1.56i)T^{2} \) |
| 5 | \( 1 + (-0.419 + 0.389i)T + (0.373 - 4.98i)T^{2} \) |
| 11 | \( 1 + (-3.73 + 2.54i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (4.93 - 3.36i)T + (4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-1.91 - 0.288i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-2.20 + 3.82i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.88 - 4.79i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (9.79 + 1.47i)T + (27.7 + 8.54i)T^{2} \) |
| 31 | \( 1 - 4.16T + 31T^{2} \) |
| 37 | \( 1 + (-0.224 - 0.573i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (-0.993 - 0.306i)T + (33.8 + 23.0i)T^{2} \) |
| 43 | \( 1 + (-11.2 + 3.48i)T + (35.5 - 24.2i)T^{2} \) |
| 47 | \( 1 + (-3.09 - 1.49i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (-3.39 + 8.64i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (-0.182 + 0.799i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (-5.54 + 6.95i)T + (-13.5 - 59.4i)T^{2} \) |
| 67 | \( 1 - 4.43T + 67T^{2} \) |
| 71 | \( 1 + (4.84 + 6.07i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-5.43 - 3.70i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + (-2.38 - 1.62i)T + (30.3 + 77.2i)T^{2} \) |
| 89 | \( 1 + (0.899 - 12.0i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + (4.39 + 7.61i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.20582838004986149864842382740, −9.765619794469968358891695865286, −9.349951436811152303497111365525, −8.894568247100093584917155875813, −7.86868460065913416240517768507, −7.14320905277154574629811901908, −5.35362547648514841967542954669, −3.82119693582714591247593877201, −2.56116504924484791158224920835, −1.61463439691237597157240537232,
1.05497551669610892095116589422, 2.30284844278491324362770270957, 4.16440322599318146709545023051, 5.89132255467542721808876130936, 7.07043454107003919371277419784, 7.50119805332830887485801042568, 8.168764531387782625161487291465, 9.227837266987519990453645409259, 9.935590252138363998608119389397, 10.45175142194352823510188101559
