L(s) = 1 | + (−0.5 + 0.866i)2-s + (1.5 − 2.59i)3-s + (0.500 + 0.866i)4-s + 3·5-s + (1.5 + 2.59i)6-s − 3·8-s + (−3 − 5.19i)9-s + (−1.5 + 2.59i)10-s + (1.5 − 2.59i)11-s + 3·12-s + (−1 − 3.46i)13-s + (4.5 − 7.79i)15-s + (0.500 − 0.866i)16-s + (1 + 1.73i)17-s + 6·18-s + (0.5 + 0.866i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.866 − 1.49i)3-s + (0.250 + 0.433i)4-s + 1.34·5-s + (0.612 + 1.06i)6-s − 1.06·8-s + (−1 − 1.73i)9-s + (−0.474 + 0.821i)10-s + (0.452 − 0.783i)11-s + 0.866·12-s + (−0.277 − 0.960i)13-s + (1.16 − 2.01i)15-s + (0.125 − 0.216i)16-s + (0.242 + 0.420i)17-s + 1.41·18-s + (0.114 + 0.198i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.02908 - 0.529771i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.02908 - 0.529771i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (1 + 3.46i)T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.5 + 2.59i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 - 3T + 5T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.5 - 6.06i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.5 - 6.06i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.5 + 11.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 13T + 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.27962439364336452354976674814, −9.188914032787288603776484306400, −8.598070929724526489489679568355, −7.84105277319775683298564600351, −7.07219297765751453787319442940, −6.21532568445604949993832185577, −5.68406590572197503629335054721, −3.32499173102226489993322310894, −2.54814639524729726420235275367, −1.31299646496539755696952707505,
1.91499391943724190431809738986, 2.60732321400380514698232162966, 3.95362846861699277713048297047, 5.01777702992843472652753311492, 5.92365999219625160161204773627, 7.09393507978738512689455294188, 8.699385849675814534459482604763, 9.324793331059242290795139831263, 9.842038728476668612465377262783, 10.16951030776102695258004177625