L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.939 + 0.342i)3-s + (−0.499 + 0.866i)4-s + (0.173 + 0.984i)5-s + (−0.173 − 0.984i)6-s + 0.999·8-s + (0.766 + 0.642i)9-s + (0.766 − 0.642i)10-s + (−0.766 + 0.642i)12-s + (−0.173 + 0.984i)15-s + (−0.5 − 0.866i)16-s + (−1.17 + 0.984i)17-s + (0.173 − 0.984i)18-s + (0.939 + 0.342i)19-s + (−0.939 − 0.342i)20-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.939 + 0.342i)3-s + (−0.499 + 0.866i)4-s + (0.173 + 0.984i)5-s + (−0.173 − 0.984i)6-s + 0.999·8-s + (0.766 + 0.642i)9-s + (0.766 − 0.642i)10-s + (−0.766 + 0.642i)12-s + (−0.173 + 0.984i)15-s + (−0.5 − 0.866i)16-s + (−1.17 + 0.984i)17-s + (0.173 − 0.984i)18-s + (0.939 + 0.342i)19-s + (−0.939 − 0.342i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.079488722\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.079488722\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.939 - 0.342i)T \) |
| 5 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 + (-0.939 - 0.342i)T \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (1.70 + 0.300i)T + (0.939 + 0.342i)T^{2} \) |
| 29 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 31 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (-1.26 + 1.50i)T + (-0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 + (-1.26 - 0.223i)T + (0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 83 | \( 1 + (0.592 + 0.342i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (0.173 - 0.984i)T^{2} \) |
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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
−10.10719719432316693341343589360, −9.432549718038559376929704426150, −8.501934287237235308208006988022, −7.86458492934210189055543142924, −7.06379303665838597026347890021, −5.88345817398486954586853645900, −4.27470047892366237820548221765, −3.76138377013397021398812762999, −2.61558531307869596695200374809, −1.96340656671429912338059742902,
1.14097235289028070002378344405, 2.41427153936614947241901769881, 4.04557969947823059626660792730, 4.86973238751032736335935348781, 5.86835416279912217717722928687, 6.89782064530354226564707152273, 7.56435916205528844172187204456, 8.457249364686240513403170595288, 8.913528411163076398391815166062, 9.627588562921791253330842425050