| L(s) = 1 | + (0.309 + 0.951i)2-s + (1.09 − 0.792i)3-s + (−0.809 + 0.587i)4-s + (−2.15 − 0.587i)5-s + (1.09 + 0.792i)6-s − 0.833·7-s + (−0.809 − 0.587i)8-s + (−0.365 + 1.12i)9-s + (−0.107 − 2.23i)10-s + (0.257 + 0.792i)11-s + (−0.416 + 1.28i)12-s + (1.41 − 4.34i)13-s + (−0.257 − 0.792i)14-s + (−2.81 + 1.06i)15-s + (0.309 − 0.951i)16-s + (−4.41 − 3.20i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.629 − 0.457i)3-s + (−0.404 + 0.293i)4-s + (−0.964 − 0.262i)5-s + (0.445 + 0.323i)6-s − 0.314·7-s + (−0.286 − 0.207i)8-s + (−0.121 + 0.374i)9-s + (−0.0340 − 0.706i)10-s + (0.0776 + 0.238i)11-s + (−0.120 + 0.370i)12-s + (0.391 − 1.20i)13-s + (−0.0688 − 0.211i)14-s + (−0.727 + 0.275i)15-s + (0.0772 − 0.237i)16-s + (−1.06 − 0.777i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.884801 + 0.213399i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.884801 + 0.213399i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.15 + 0.587i)T \) |
| good | 3 | \( 1 + (-1.09 + 0.792i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + 0.833T + 7T^{2} \) |
| 11 | \( 1 + (-0.257 - 0.792i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.41 + 4.34i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (4.41 + 3.20i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-7.00 - 5.08i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.09 - 3.35i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (2.64 - 1.92i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (4.85 + 3.52i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.26 + 6.95i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.576 + 1.77i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 1.63T + 43T^{2} \) |
| 47 | \( 1 + (-0.674 + 0.489i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (5.19 - 3.77i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-4.18 + 12.8i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.81 - 5.59i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (1.21 + 0.881i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-1.91 + 1.38i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.02 + 3.16i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (4.18 - 3.03i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-9.97 - 7.25i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (2.16 + 6.66i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (8.97 - 6.51i)T + (29.9 - 92.2i)T^{2} \) |
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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
−15.71722160923512237471653514967, −14.59849972747249108909689432482, −13.45810619216296466924212719357, −12.60639995224576316036787102613, −11.19477810397215719145609273934, −9.299343603771264096174189573705, −7.995955762103588709137229109224, −7.31881562226818636948363357107, −5.34535994999870021536395595042, −3.44266246576126949283524513030,
3.17459120572987956606991684069, 4.36420157987861449824327394935, 6.71674076957440159812442047772, 8.573252849835968570618815211044, 9.475582019359727795610839772923, 11.04574701177117349738944016356, 11.83181206204205240403486297782, 13.26576119433431699757004728734, 14.40079464679263799472507203816, 15.34091117881981262133671140841
