| L(s) = 1 | + (0.374 + 0.240i)3-s + (−0.415 + 0.909i)5-s + (−4.12 − 1.21i)7-s + (−1.16 − 2.54i)9-s + (−1.13 + 1.31i)11-s + (−6.87 + 2.01i)13-s + (−0.374 + 0.240i)15-s + (1.05 + 7.34i)17-s + (0.518 − 3.60i)19-s + (−1.25 − 1.44i)21-s + (2.42 − 4.13i)23-s + (−0.654 − 0.755i)25-s + (0.367 − 2.55i)27-s + (−0.881 − 6.13i)29-s + (−4.27 + 2.74i)31-s + ⋯ |
| L(s) = 1 | + (0.216 + 0.139i)3-s + (−0.185 + 0.406i)5-s + (−1.56 − 0.458i)7-s + (−0.387 − 0.849i)9-s + (−0.342 + 0.395i)11-s + (−1.90 + 0.559i)13-s + (−0.0967 + 0.0621i)15-s + (0.256 + 1.78i)17-s + (0.119 − 0.828i)19-s + (−0.273 − 0.316i)21-s + (0.505 − 0.862i)23-s + (−0.130 − 0.151i)25-s + (0.0707 − 0.492i)27-s + (−0.163 − 1.13i)29-s + (−0.767 + 0.492i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00132495 - 0.0502435i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00132495 - 0.0502435i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (-2.42 + 4.13i)T \) |
| good | 3 | \( 1 + (-0.374 - 0.240i)T + (1.24 + 2.72i)T^{2} \) |
| 7 | \( 1 + (4.12 + 1.21i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (1.13 - 1.31i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (6.87 - 2.01i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-1.05 - 7.34i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.518 + 3.60i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (0.881 + 6.13i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (4.27 - 2.74i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.48 - 3.25i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (2.60 - 5.69i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-3.59 - 2.31i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 2.21T + 47T^{2} \) |
| 53 | \( 1 + (7.30 + 2.14i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-1.53 + 0.450i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-3.11 + 2.00i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (5.76 + 6.65i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (6.98 + 8.05i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.93 - 13.4i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-11.4 + 3.36i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-0.267 - 0.585i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (2.38 + 1.53i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-2.51 + 5.51i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.54631082871521128877021369674, −10.32362607927184813515684334057, −9.806407019370863795489411833056, −9.053372147268119790445388812026, −7.75059279444584089289511697221, −6.79028428967966521700781367756, −6.21647037380259212914944693050, −4.62307012971785607434003055065, −3.51436487101254106124048181430, −2.57329370778867294494731004610,
0.02746086954890779333934221341, 2.52260959445038592831406117903, 3.29330649097051741522804564581, 5.09254284815095980633387327217, 5.61495422090046015629002385009, 7.19035324499860422916097207943, 7.64983550148043104699958065655, 9.012085975269315619380517466549, 9.575489427399222621150862907225, 10.44391966774966123319966343536
