| L(s) = 1 | + (−0.984 − 0.173i)2-s + (−1.69 + 0.351i)3-s + (0.939 + 0.342i)4-s + (−2.70 − 0.984i)5-s + (1.73 − 0.0521i)6-s + (2.49 − 0.866i)7-s + (−0.866 − 0.5i)8-s + (2.75 − 1.19i)9-s + (2.49 + 1.43i)10-s + (−0.289 − 0.796i)11-s + (−1.71 − 0.249i)12-s + (−0.238 + 0.655i)13-s + (−2.61 + 0.418i)14-s + (4.93 + 0.717i)15-s + (0.766 + 0.642i)16-s + (−2.72 + 4.72i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 − 0.122i)2-s + (−0.979 + 0.203i)3-s + (0.469 + 0.171i)4-s + (−1.21 − 0.440i)5-s + (0.706 − 0.0212i)6-s + (0.944 − 0.327i)7-s + (−0.306 − 0.176i)8-s + (0.917 − 0.397i)9-s + (0.788 + 0.455i)10-s + (−0.0873 − 0.240i)11-s + (−0.494 − 0.0719i)12-s + (−0.0661 + 0.181i)13-s + (−0.698 + 0.111i)14-s + (1.27 + 0.185i)15-s + (0.191 + 0.160i)16-s + (−0.662 + 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.103562 + 0.193509i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.103562 + 0.193509i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 3 | \( 1 + (1.69 - 0.351i)T \) |
| 7 | \( 1 + (-2.49 + 0.866i)T \) |
| good | 5 | \( 1 + (2.70 + 0.984i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (0.289 + 0.796i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (0.238 - 0.655i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (2.72 - 4.72i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (5.46 - 3.15i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.98 - 0.350i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.22 - 3.35i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.631 - 1.73i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + 8.28T + 37T^{2} \) |
| 41 | \( 1 + (-9.58 - 3.48i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.22 - 6.92i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (7.93 - 2.88i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-4.75 + 2.74i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.28 - 1.91i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.396 - 1.08i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.33 - 7.55i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (10.1 - 5.87i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 5.01iT - 73T^{2} \) |
| 79 | \( 1 + (-1.54 + 8.74i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.72 + 0.990i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-0.0671 - 0.116i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (11.1 + 1.97i)T + (91.1 + 33.1i)T^{2} \) |
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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
−11.43161644494463263982830243538, −10.89280403827870167069006548167, −10.14924953573189513578632076440, −8.676640771260171846359359289097, −8.121817964481500969658301034404, −7.11727638497273947084776826952, −6.00993863477769297828283162997, −4.61155406344155185047092979173, −3.91152532346648043429581158273, −1.51620165284933400280766202776,
0.21327153955931961779426859729, 2.23547284113220574532433695287, 4.20957715164693944854925321479, 5.18975897452023469402288114586, 6.54486765678748152888935564728, 7.34358984899662592250389424642, 8.060364871947409578558612263243, 9.117493088327705692445688024442, 10.49569049897390259216688404133, 11.05658589241625553893411740037
