| L(s) = 1 | + (−2.40 − 1.79i)3-s − 2.23i·5-s + 10.2·7-s + (2.53 + 8.63i)9-s + 8.19i·11-s + 13.5·13-s + (−4.02 + 5.36i)15-s + 15.4i·17-s − 25.4·19-s + (−24.5 − 18.3i)21-s + 17.9i·23-s − 5.00·25-s + (9.44 − 25.2i)27-s + 42.0i·29-s − 38.4·31-s + ⋯ |
| L(s) = 1 | + (−0.800 − 0.599i)3-s − 0.447i·5-s + 1.45·7-s + (0.281 + 0.959i)9-s + 0.744i·11-s + 1.04·13-s + (−0.268 + 0.357i)15-s + 0.910i·17-s − 1.34·19-s + (−1.16 − 0.874i)21-s + 0.778i·23-s − 0.200·25-s + (0.349 − 0.936i)27-s + 1.44i·29-s − 1.24·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.357859599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.357859599\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.40 + 1.79i)T \) |
| 5 | \( 1 + 2.23iT \) |
| good | 7 | \( 1 - 10.2T + 49T^{2} \) |
| 11 | \( 1 - 8.19iT - 121T^{2} \) |
| 13 | \( 1 - 13.5T + 169T^{2} \) |
| 17 | \( 1 - 15.4iT - 289T^{2} \) |
| 19 | \( 1 + 25.4T + 361T^{2} \) |
| 23 | \( 1 - 17.9iT - 529T^{2} \) |
| 29 | \( 1 - 42.0iT - 841T^{2} \) |
| 31 | \( 1 + 38.4T + 961T^{2} \) |
| 37 | \( 1 + 11.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 46.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 54.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 43.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 82.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 45.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 93.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 34.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 68.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 44.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 11.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 144. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 63.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 63.9T + 9.40e3T^{2} \) |
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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
−10.25381581537781296057125428065, −8.826461125828569432085782778921, −8.274090700887095480698816723206, −7.44509681642269079803234819020, −6.52590811785008789238060008195, −5.54584834107296520823428172746, −4.85106684005577961296801195910, −3.92992262559809397192969474095, −1.88022969817211649777293691101, −1.37857402159267403906595163841,
0.50535040380761754001450683914, 2.01154852746216825360464602513, 3.57825257643865041598062550748, 4.43991987818660862541268509908, 5.34099626632700522055706318500, 6.15071074670008154416194215235, 7.01416857741814806386326017301, 8.236156969566155836525918821919, 8.750625553707540034890658762790, 9.919557931321692109013425232173
