| L(s) = 1 | + (−0.120 + 2.11i)2-s + (2.08 + 0.677i)3-s + (−2.47 − 0.284i)4-s + (−2.23 + 0.0274i)5-s + (−1.68 + 4.33i)6-s + (−0.239 + 0.566i)7-s + (0.177 − 1.02i)8-s + (1.46 + 1.06i)9-s + (0.212 − 4.73i)10-s + (−2.87 + 1.64i)11-s + (−4.97 − 2.26i)12-s + (2.04 + 3.61i)13-s + (−1.16 − 0.575i)14-s + (−4.68 − 1.45i)15-s + (−2.70 − 0.628i)16-s + (−0.209 − 0.586i)17-s + ⋯ |
| L(s) = 1 | + (−0.0855 + 1.49i)2-s + (1.20 + 0.391i)3-s + (−1.23 − 0.142i)4-s + (−0.999 + 0.0122i)5-s + (−0.688 + 1.76i)6-s + (−0.0904 + 0.214i)7-s + (0.0628 − 0.363i)8-s + (0.487 + 0.354i)9-s + (0.0672 − 1.49i)10-s + (−0.868 + 0.496i)11-s + (−1.43 − 0.655i)12-s + (0.566 + 1.00i)13-s + (−0.312 − 0.153i)14-s + (−1.20 − 0.376i)15-s + (−0.675 − 0.157i)16-s + (−0.0507 − 0.142i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.338330 - 1.28285i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.338330 - 1.28285i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.23 - 0.0274i)T \) |
| 11 | \( 1 + (2.87 - 1.64i)T \) |
| good | 2 | \( 1 + (0.120 - 2.11i)T + (-1.98 - 0.227i)T^{2} \) |
| 3 | \( 1 + (-2.08 - 0.677i)T + (2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + (0.239 - 0.566i)T + (-4.87 - 5.01i)T^{2} \) |
| 13 | \( 1 + (-2.04 - 3.61i)T + (-6.71 + 11.1i)T^{2} \) |
| 17 | \( 1 + (0.209 + 0.586i)T + (-13.1 + 10.7i)T^{2} \) |
| 19 | \( 1 + (0.0830 - 2.90i)T + (-18.9 - 1.08i)T^{2} \) |
| 23 | \( 1 + (0.274 - 0.237i)T + (3.27 - 22.7i)T^{2} \) |
| 29 | \( 1 + (0.438 + 0.640i)T + (-10.5 + 27.0i)T^{2} \) |
| 31 | \( 1 + (-0.558 + 2.75i)T + (-28.5 - 12.0i)T^{2} \) |
| 37 | \( 1 + (0.544 + 0.593i)T + (-3.16 + 36.8i)T^{2} \) |
| 41 | \( 1 + (-1.21 - 4.60i)T + (-35.7 + 20.1i)T^{2} \) |
| 43 | \( 1 + (2.12 - 0.306i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + (-0.349 - 0.427i)T + (-9.33 + 46.0i)T^{2} \) |
| 53 | \( 1 + (-2.11 - 9.10i)T + (-47.5 + 23.3i)T^{2} \) |
| 59 | \( 1 + (0.0374 - 0.142i)T + (-51.3 - 29.0i)T^{2} \) |
| 61 | \( 1 + (-8.25 + 0.471i)T + (60.6 - 6.95i)T^{2} \) |
| 67 | \( 1 + (-3.08 - 4.80i)T + (-27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (6.92 - 8.98i)T + (-18.0 - 68.6i)T^{2} \) |
| 73 | \( 1 + (-11.0 - 6.64i)T + (34.0 + 64.5i)T^{2} \) |
| 79 | \( 1 + (-4.92 + 5.06i)T + (-2.25 - 78.9i)T^{2} \) |
| 83 | \( 1 + (-16.4 - 1.41i)T + (81.7 + 14.1i)T^{2} \) |
| 89 | \( 1 + (6.52 + 1.91i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-2.30 - 1.21i)T + (54.7 + 80.0i)T^{2} \) |
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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
−11.09690255547931067268148110531, −9.825696062111341721256892037654, −8.997251217205636524045282437176, −8.330663737833918780482572376329, −7.75650510667175480368081242475, −6.98384705398954043258451817864, −5.87632407602210086078339597439, −4.63617687293267116239469586488, −3.83791045299791833866470163878, −2.50807674459219248180484287927,
0.64769812432110819654849965790, 2.27480376198227178702341138572, 3.22557637247938266653306359293, 3.71197091409561514844981825217, 5.11821268728347918493297388826, 6.87202561142633465516806231931, 7.934388171618970377535059935143, 8.455296560468794768531451075002, 9.255070369921122902149674696120, 10.47337079540042180905695555280
