L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (1.35 − 2.33i)5-s − 0.999i·8-s − 2.70i·10-s + (0.205 − 0.118i)11-s + (−2.31 − 1.33i)13-s + (−0.5 − 0.866i)16-s − 5.86·17-s − 1.15i·19-s + (−1.35 − 2.33i)20-s + (0.118 − 0.205i)22-s + (−7.02 − 4.05i)23-s + (−1.14 − 1.98i)25-s − 2.67·26-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.603 − 1.04i)5-s − 0.353i·8-s − 0.853i·10-s + (0.0621 − 0.0358i)11-s + (−0.641 − 0.370i)13-s + (−0.125 − 0.216i)16-s − 1.42·17-s − 0.264i·19-s + (−0.301 − 0.522i)20-s + (0.0253 − 0.0439i)22-s + (−1.46 − 0.845i)23-s + (−0.229 − 0.397i)25-s − 0.524·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.708397376\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.708397376\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.35 + 2.33i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.205 + 0.118i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.31 + 1.33i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 5.86T + 17T^{2} \) |
| 19 | \( 1 + 1.15iT - 19T^{2} \) |
| 23 | \( 1 + (7.02 + 4.05i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (8.88 - 5.13i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5.21 + 3.01i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 9.68T + 37T^{2} \) |
| 41 | \( 1 + (-3.81 + 6.61i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.69 - 4.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.221 + 0.383i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 0.219iT - 53T^{2} \) |
| 59 | \( 1 + (-0.983 + 1.70i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.8 + 6.27i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.48 + 7.76i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2.24iT - 71T^{2} \) |
| 73 | \( 1 + 7.25iT - 73T^{2} \) |
| 79 | \( 1 + (-5.43 - 9.41i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.762 - 1.32i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + (-1.37 + 0.795i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.646958082882134237150209404078, −7.74607121967323758439141671436, −6.81938679630081203174092338286, −5.91549470446777031153893622088, −5.33324304987444767423766572639, −4.51950730307621505478490446620, −3.85964693604926731292693034233, −2.46799872780094276176387563871, −1.83573266925211743740850978111, −0.39489255962955160571110399887,
2.01829714711857107566878206675, 2.53477512670747447207736222024, 3.76784014072112483278574250341, 4.38316584833451383576265035363, 5.60262952537868511183766861350, 6.06815989184725578895269749039, 6.90353164173719634287652188424, 7.42701315115904140077112345766, 8.285041993327312445850724201324, 9.410219601862241537244515241863