L(s) = 1 | − 1.73i·2-s − 0.999·4-s − 3.46i·5-s + 2·7-s − 1.73i·8-s − 5.99·10-s + 6·11-s − 3.46i·14-s − 5·16-s − 3·17-s + 4·19-s + 3.46i·20-s − 10.3i·22-s + 6·23-s − 6.99·25-s + ⋯ |
L(s) = 1 | − 1.22i·2-s − 0.499·4-s − 1.54i·5-s + 0.755·7-s − 0.612i·8-s − 1.89·10-s + 1.80·11-s − 0.925i·14-s − 1.25·16-s − 0.727·17-s + 0.917·19-s + 0.774i·20-s − 2.21i·22-s + 1.25·23-s − 1.39·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.352889 - 1.89923i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.352889 - 1.89923i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 + (-2 - 5.19i)T \) |
good | 2 | \( 1 + 1.73iT - 2T^{2} \) |
| 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 5.19iT - 43T^{2} \) |
| 47 | \( 1 - 5.19iT - 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 - 1.73iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 14T + 67T^{2} \) |
| 71 | \( 1 + 1.73iT - 71T^{2} \) |
| 73 | \( 1 + 10.3iT - 73T^{2} \) |
| 79 | \( 1 - 15.5iT - 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 - 11T + 97T^{2} \) |
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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
−9.645096010915611562991231968182, −9.201509492856631165075593383078, −8.581271984992672113834928231185, −7.35974939544665156116596977847, −6.27137472878734127817790021153, −4.90382737578548711706686072427, −4.37290338351873368521713976011, −3.27424464561895398607380194715, −1.61666644346413186719296289506, −1.12103771812187319926668138464,
1.91912924591334761207820451529, 3.27920306185655880357948539361, 4.44845375668621941487040205351, 5.64765497079532599249654896213, 6.50035462255486572978613942605, 7.05491958347309582255858622265, 7.64138108027541638058023826948, 8.767674385179784012514615145752, 9.476605558325649063804558491292, 10.73552018675074430738703220946