| L(s) = 1 | + (−0.104 + 0.994i)2-s + (−0.669 − 0.743i)3-s + (−0.978 − 0.207i)4-s + (0.363 + 0.161i)5-s + (0.809 − 0.587i)6-s + (−1.41 − 2.23i)7-s + (0.309 − 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.198 + 0.344i)10-s + (−0.820 + 3.21i)11-s + (0.5 + 0.866i)12-s + (−2.60 − 1.89i)13-s + (2.37 − 1.17i)14-s + (−0.122 − 0.378i)15-s + (0.913 + 0.406i)16-s + (−0.654 − 6.22i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 + 0.703i)2-s + (−0.386 − 0.429i)3-s + (−0.489 − 0.103i)4-s + (0.162 + 0.0723i)5-s + (0.330 − 0.239i)6-s + (−0.535 − 0.844i)7-s + (0.109 − 0.336i)8-s + (−0.0348 + 0.331i)9-s + (−0.0628 + 0.108i)10-s + (−0.247 + 0.968i)11-s + (0.144 + 0.249i)12-s + (−0.722 − 0.525i)13-s + (0.633 − 0.313i)14-s + (−0.0317 − 0.0976i)15-s + (0.228 + 0.101i)16-s + (−0.158 − 1.51i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.346 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.260703 - 0.374022i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.260703 - 0.374022i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (0.669 + 0.743i)T \) |
| 7 | \( 1 + (1.41 + 2.23i)T \) |
| 11 | \( 1 + (0.820 - 3.21i)T \) |
| good | 5 | \( 1 + (-0.363 - 0.161i)T + (3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (2.60 + 1.89i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.654 + 6.22i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (4.92 - 1.04i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (3.00 + 5.21i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.61 - 4.95i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.289 - 0.128i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-2.18 + 2.42i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (-2.86 + 8.80i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 2.62T + 43T^{2} \) |
| 47 | \( 1 + (7.69 - 1.63i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-10.6 + 4.74i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-7.74 - 1.64i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (3.35 + 1.49i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (7.15 - 12.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.67 - 3.39i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (8.02 + 1.70i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-0.0695 + 0.662i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-7.09 + 5.15i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.97 - 5.15i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.7 - 9.23i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.35572917250743189739434907243, −10.12243176981458839084849407826, −8.887366683087826410671145343127, −7.69065784444916340756031943400, −7.08313469981839749948024241978, −6.32984101994090345081266914274, −5.12028233731250270864111652299, −4.20737543439755257169315210501, −2.42588746818709385311346460746, −0.28543454716528358712941897044,
2.01748695891146912723983979175, 3.34182798241863597063382309163, 4.43288712459619203853701298258, 5.70476823424427643335561997603, 6.31886011285844763213002673040, 7.999765410250553123341603714399, 8.866531493417312437939500680252, 9.685599463235096275145382878524, 10.40091028588899319829011028187, 11.40992984197977350862589785143
