| L(s) = 1 | + (1.46 + 0.924i)3-s + (−0.217 + 1.23i)5-s + (2.00 − 1.68i)7-s + (1.29 + 2.70i)9-s + (−0.0755 − 0.428i)11-s + (0.140 − 0.0512i)13-s + (−1.45 + 1.60i)15-s + (2.14 − 3.72i)17-s + (3.46 + 5.99i)19-s + (4.49 − 0.612i)21-s + (1.58 + 1.32i)23-s + (3.22 + 1.17i)25-s + (−0.610 + 5.16i)27-s + (−8.62 − 3.13i)29-s + (0.439 + 0.368i)31-s + ⋯ |
| L(s) = 1 | + (0.845 + 0.533i)3-s + (−0.0972 + 0.551i)5-s + (0.758 − 0.636i)7-s + (0.430 + 0.902i)9-s + (−0.0227 − 0.129i)11-s + (0.0390 − 0.0142i)13-s + (−0.376 + 0.414i)15-s + (0.521 − 0.902i)17-s + (0.793 + 1.37i)19-s + (0.981 − 0.133i)21-s + (0.329 + 0.276i)23-s + (0.645 + 0.234i)25-s + (−0.117 + 0.993i)27-s + (−1.60 − 0.582i)29-s + (0.0789 + 0.0662i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.09687 + 0.902703i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09687 + 0.902703i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.46 - 0.924i)T \) |
| good | 5 | \( 1 + (0.217 - 1.23i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-2.00 + 1.68i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.0755 + 0.428i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-0.140 + 0.0512i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.14 + 3.72i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.46 - 5.99i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.58 - 1.32i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (8.62 + 3.13i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.439 - 0.368i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-1.47 + 2.55i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.70 - 1.71i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.773 - 4.38i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.17 + 4.34i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 7.31T + 53T^{2} \) |
| 59 | \( 1 + (1.82 - 10.3i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-6.60 + 5.54i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-3.27 + 1.19i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (1.25 - 2.18i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.40 + 9.36i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (13.9 + 5.09i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.90 - 1.42i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (2.90 + 5.03i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.44 + 13.8i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.15887912111038136029630634050, −9.551783138685112496247886272741, −8.568544014295286641881434305377, −7.54214862185703890692253003338, −7.37928234529493773710810989319, −5.76742268116838158624125737607, −4.79692591090502184708359160489, −3.76877500548257546731725570538, −2.98630245240311720238464808415, −1.56566623204040043139883936002,
1.21384336856715148129265263123, 2.34295910299631314678176515497, 3.49936830114154830915345308394, 4.71504581328536520758219721968, 5.58990880628129552501109113690, 6.82575845892608323849778172309, 7.62926676224100824040485182602, 8.501827963175039273108537931828, 8.938592497931734700047032604626, 9.799599461360033461204219873339
