| L(s) = 1 | + (0.104 + 0.994i)2-s + (−0.175 − 0.195i)3-s + (−0.978 + 0.207i)4-s + (1.83 − 1.27i)5-s + (0.175 − 0.195i)6-s − 2.84·7-s + (−0.309 − 0.951i)8-s + (0.306 − 2.91i)9-s + (1.46 + 1.69i)10-s + (0.311 − 0.226i)11-s + (0.212 + 0.154i)12-s + (−0.375 + 3.57i)13-s + (−0.297 − 2.83i)14-s + (−0.571 − 0.133i)15-s + (0.913 − 0.406i)16-s + (−0.960 − 0.204i)17-s + ⋯ |
| L(s) = 1 | + (0.0739 + 0.703i)2-s + (−0.101 − 0.112i)3-s + (−0.489 + 0.103i)4-s + (0.820 − 0.571i)5-s + (0.0717 − 0.0796i)6-s − 1.07·7-s + (−0.109 − 0.336i)8-s + (0.102 − 0.971i)9-s + (0.462 + 0.534i)10-s + (0.0939 − 0.0682i)11-s + (0.0613 + 0.0445i)12-s + (−0.104 + 0.990i)13-s + (−0.0796 − 0.757i)14-s + (−0.147 − 0.0344i)15-s + (0.228 − 0.101i)16-s + (−0.232 − 0.0495i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.375 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.910208 - 0.613639i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.910208 - 0.613639i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.104 - 0.994i)T \) |
| 5 | \( 1 + (-1.83 + 1.27i)T \) |
| 19 | \( 1 + (-0.541 + 4.32i)T \) |
| good | 3 | \( 1 + (0.175 + 0.195i)T + (-0.313 + 2.98i)T^{2} \) |
| 7 | \( 1 + 2.84T + 7T^{2} \) |
| 11 | \( 1 + (-0.311 + 0.226i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.375 - 3.57i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (0.960 + 0.204i)T + (15.5 + 6.91i)T^{2} \) |
| 23 | \( 1 + (8.05 + 3.58i)T + (15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (-9.74 + 2.07i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (1.68 + 5.17i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (6.52 + 4.74i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (5.29 - 2.35i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-4.40 + 7.63i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.07 - 0.866i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-9.32 + 1.98i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (3.17 - 1.41i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-0.266 - 0.118i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-7.31 + 8.12i)T + (-7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (-0.0465 - 0.0517i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-0.956 - 9.10i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (1.79 + 1.99i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.609 - 1.87i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-9.78 - 4.35i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (-7.05 - 7.84i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−9.648996151116086134719355070481, −9.094296346431421510102479813983, −8.377361483778416747543577829159, −6.93064942010326903796210354126, −6.49767641044458964917461791564, −5.83163753441236007316156278387, −4.65625808154277554107931985271, −3.75052685140173996098588644004, −2.28994242628305204582634314704, −0.49361783681905572040294613416,
1.68269816193834458379804130953, 2.81380594765206226176612445349, 3.60123808963740823864468336898, 4.98473496100625599503188406427, 5.81381363436667088435894903179, 6.62122385910177879659299286360, 7.76076429742874774452233651985, 8.677269206101566752859993067368, 9.920484719316955480592565327882, 10.12118489010225219019234269790
