| L(s) = 1 | + (−0.984 + 0.173i)2-s + (0.754 − 1.55i)3-s + (0.939 − 0.342i)4-s + (−0.486 − 0.408i)5-s + (−0.472 + 1.66i)6-s + (−2.18 + 1.48i)7-s + (−0.866 + 0.5i)8-s + (−1.86 − 2.35i)9-s + (0.550 + 0.317i)10-s + (−3.72 − 4.43i)11-s + (0.175 − 1.72i)12-s + (0.505 + 0.0891i)13-s + (1.89 − 1.84i)14-s + (−1.00 + 0.450i)15-s + (0.766 − 0.642i)16-s + (1.66 − 2.88i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (0.435 − 0.900i)3-s + (0.469 − 0.171i)4-s + (−0.217 − 0.182i)5-s + (−0.192 + 0.680i)6-s + (−0.826 + 0.562i)7-s + (−0.306 + 0.176i)8-s + (−0.620 − 0.784i)9-s + (0.174 + 0.100i)10-s + (−1.12 − 1.33i)11-s + (0.0507 − 0.497i)12-s + (0.140 + 0.0247i)13-s + (0.506 − 0.493i)14-s + (−0.259 + 0.116i)15-s + (0.191 − 0.160i)16-s + (0.404 − 0.700i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.149211 - 0.555374i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.149211 - 0.555374i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (-0.754 + 1.55i)T \) |
| 7 | \( 1 + (2.18 - 1.48i)T \) |
| good | 5 | \( 1 + (0.486 + 0.408i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (3.72 + 4.43i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.505 - 0.0891i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.66 + 2.88i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.56 - 2.05i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.488 + 1.34i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (3.07 - 0.541i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.46 - 6.77i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (0.189 - 0.328i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.877 + 4.97i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.02 + 5.89i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (5.23 + 1.90i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 3.85iT - 53T^{2} \) |
| 59 | \( 1 + (-2.85 - 2.39i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.871 - 2.39i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.36 - 7.75i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (2.67 + 1.54i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-11.5 + 6.66i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.238 + 1.35i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.33 - 7.57i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-2.08 - 3.61i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (11.4 + 13.5i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−10.91960702238197362127835789563, −9.956403098389953037913483078626, −8.736037361780162030546300486454, −8.412928813212897491371878351391, −7.36974926227408522083173531600, −6.32754506816499534768474368311, −5.54600788054892969760214491236, −3.37431151770909329781600527398, −2.38639766401936764598854497665, −0.42517260809805096198502751196,
2.34519716176487821414055159290, 3.55122996781532694581381853150, 4.63795223705386414232150462485, 6.08845784605018474943204361192, 7.42980242470428183225055349285, 7.998458988168337715772537285597, 9.309346076374019168826962794169, 9.868535939140699986501482691091, 10.57387346223830353010715993915, 11.31253895365970928597379951252
