| L(s) = 1 | + (0.309 + 0.951i)2-s + (1.47 − 1.07i)3-s + (−0.809 + 0.587i)4-s + (1.11 − 1.93i)5-s + (1.47 + 1.07i)6-s + 7-s + (−0.809 − 0.587i)8-s + (0.104 − 0.321i)9-s + (2.18 + 0.464i)10-s + (−0.284 − 0.875i)11-s + (−0.564 + 1.73i)12-s + (0.486 − 1.49i)13-s + (0.309 + 0.951i)14-s + (−0.427 − 4.06i)15-s + (0.309 − 0.951i)16-s + (3.08 + 2.23i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.853 − 0.620i)3-s + (−0.404 + 0.293i)4-s + (0.499 − 0.866i)5-s + (0.603 + 0.438i)6-s + 0.377·7-s + (−0.286 − 0.207i)8-s + (0.0348 − 0.107i)9-s + (0.691 + 0.147i)10-s + (−0.0857 − 0.263i)11-s + (−0.162 + 0.501i)12-s + (0.134 − 0.415i)13-s + (0.0825 + 0.254i)14-s + (−0.110 − 1.04i)15-s + (0.0772 − 0.237i)16-s + (0.747 + 0.543i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.01487 - 0.0422055i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.01487 - 0.0422055i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + (-1.11 + 1.93i)T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + (-1.47 + 1.07i)T + (0.927 - 2.85i)T^{2} \) |
| 11 | \( 1 + (0.284 + 0.875i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.486 + 1.49i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-3.08 - 2.23i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.83 + 1.33i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.0315 + 0.0971i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (2.01 - 1.46i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-3.20 - 2.33i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.656 - 2.01i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (2.69 - 8.29i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 7.79T + 43T^{2} \) |
| 47 | \( 1 + (5.68 - 4.12i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.315 + 0.228i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.101 + 0.313i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.09 + 6.43i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-0.132 - 0.0962i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (3.87 - 2.81i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.83 - 11.7i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-8.66 + 6.29i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (12.5 + 9.10i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (3.07 + 9.46i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-7.66 + 5.56i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.74440369767403066607830861914, −10.37688992749001284619416568320, −9.286357680945438552398626296766, −8.231873394896592070885573541770, −8.108175388595447669952600858179, −6.73695244024724284193001340495, −5.61721076352969658613593299576, −4.68167955671580556883965320169, −3.12658237187100748757173901353, −1.55580622744009536859844729727,
2.04477253170785379228223072833, 3.14687240495120254872102035976, 4.08284927434453089181097596709, 5.40133005811194615877789120880, 6.65322217801187417690850385195, 7.932333046046691905139683522755, 9.005964828225553402367119990550, 9.810823747015664320296796713545, 10.39895155877188630645988691422, 11.41908826817140627541742863984
