| L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (3.75 − 1.36i)5-s + (−0.766 + 0.642i)6-s + (−1.5 − 2.59i)7-s + (−0.5 + 0.866i)8-s + (−0.347 − 1.96i)9-s + (−0.694 − 3.93i)10-s + (−1 + 1.73i)11-s + (0.499 + 0.866i)12-s + (−0.766 + 0.642i)13-s + (−2.81 + 1.02i)14-s + (−3.75 − 1.36i)15-s + (0.766 + 0.642i)16-s + (0.520 − 2.95i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.442 − 0.371i)3-s + (−0.469 − 0.171i)4-s + (1.68 − 0.611i)5-s + (−0.312 + 0.262i)6-s + (−0.566 − 0.981i)7-s + (−0.176 + 0.306i)8-s + (−0.115 − 0.656i)9-s + (−0.219 − 1.24i)10-s + (−0.301 + 0.522i)11-s + (0.144 + 0.249i)12-s + (−0.212 + 0.178i)13-s + (−0.753 + 0.274i)14-s + (−0.970 − 0.353i)15-s + (0.191 + 0.160i)16-s + (0.126 − 0.716i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.228499 - 1.42151i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.228499 - 1.42151i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (0.766 + 0.642i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (-3.75 + 1.36i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.766 - 0.642i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.520 + 2.95i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.939 - 0.342i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.868 + 4.92i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (6.12 + 5.14i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.75 - 1.36i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.38 - 7.87i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.939 - 0.342i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-2.60 + 14.7i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (1.87 + 0.684i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.520 - 2.95i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (1.87 - 0.684i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-6.89 - 5.78i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (7.66 + 6.42i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (0.347 - 1.96i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−9.848811103328479000936273805762, −9.640212530710415225720853058150, −8.617882443786692971814160266504, −7.10954215142039244996352490283, −6.41031740365424846463615354307, −5.45040992262970050496267113057, −4.62154498439553498294574565671, −3.20759091722044004082020246190, −1.90321475318731714571285585517, −0.75306377121620432118917381904,
2.13902452911847298068460175340, 3.15708211410575663912698735661, 4.89479701157154266189784871098, 5.75862005874547180960372012380, 5.99437096407735338182436720032, 7.01330402287587724572409749010, 8.313386247174276850142095413058, 9.111039923190417535670253289874, 10.04515389661651669327061267655, 10.42471158615610268513236683313
