| L(s) = 1 | + (−0.845 − 1.13i)2-s + (−1.20 + 1.23i)3-s + (−0.570 + 1.91i)4-s + (−3.22 + 2.26i)5-s + (2.42 + 0.323i)6-s + (−3.12 + 2.62i)7-s + (2.65 − 0.974i)8-s + (−0.0723 − 2.99i)9-s + (5.29 + 1.74i)10-s + (−0.153 − 0.107i)11-s + (−1.68 − 3.02i)12-s + (3.05 + 1.42i)13-s + (5.61 + 1.32i)14-s + (1.10 − 6.73i)15-s + (−3.35 − 2.18i)16-s + (−5.49 − 3.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.597 − 0.801i)2-s + (−0.698 + 0.715i)3-s + (−0.285 + 0.958i)4-s + (−1.44 + 1.01i)5-s + (0.991 + 0.132i)6-s + (−1.18 + 0.991i)7-s + (0.938 − 0.344i)8-s + (−0.0241 − 0.999i)9-s + (1.67 + 0.552i)10-s + (−0.0463 − 0.0324i)11-s + (−0.486 − 0.873i)12-s + (0.846 + 0.394i)13-s + (1.50 + 0.354i)14-s + (0.285 − 1.73i)15-s + (−0.837 − 0.546i)16-s + (−1.33 − 0.769i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0122508 - 0.0213110i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0122508 - 0.0213110i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.845 + 1.13i)T \) |
| 3 | \( 1 + (1.20 - 1.23i)T \) |
| good | 5 | \( 1 + (3.22 - 2.26i)T + (1.71 - 4.69i)T^{2} \) |
| 7 | \( 1 + (3.12 - 2.62i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.153 + 0.107i)T + (3.76 + 10.3i)T^{2} \) |
| 13 | \( 1 + (-3.05 - 1.42i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (5.49 + 3.17i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.06 - 0.820i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.26 + 1.50i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-4.49 + 2.09i)T + (18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (4.77 - 5.69i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-0.525 - 1.96i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.19 + 0.435i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.0940 - 0.134i)T + (-14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (5.37 - 4.50i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (10.0 + 10.0i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.11 + 1.59i)T + (-20.1 + 55.4i)T^{2} \) |
| 61 | \( 1 + (-2.78 - 0.243i)T + (60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (2.28 - 4.89i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-5.67 - 3.27i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.42 - 2.55i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.39 + 12.0i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.658 - 1.41i)T + (-53.3 + 63.5i)T^{2} \) |
| 89 | \( 1 + (5.61 + 9.73i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.28 + 18.6i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.07645570836430746478053879669, −10.09015799855512476027593170575, −9.206474859811976563221296650368, −8.448654003788771756441584093871, −7.04370765937186507884589856045, −6.38685269394742103076798777671, −4.65935872614482139206597650850, −3.54502604972257994574258754247, −2.91756061648766335667068555896, −0.02619270902163762595247025595,
1.04794702656666903053313984361, 3.83559795988458843176953821252, 4.84456725111748002801550509715, 6.09448801316400597991374320385, 6.95131595219826258884755686057, 7.69411432256303293012927332562, 8.448963214902544383542526000871, 9.433323772615992252904510449487, 10.74209653850551582354362684394, 11.20916011863476695470022303902
