| L(s) = 1 | + (−197. + 141. i)3-s − 5.21e3i·5-s − 7.04e3·7-s + (1.91e4 − 5.58e4i)9-s + 4.61e3i·11-s + 1.29e5·13-s + (7.37e5 + 1.03e6i)15-s + 3.25e5i·17-s − 4.56e6·19-s + (1.39e6 − 9.94e5i)21-s + 7.65e6i·23-s − 1.74e7·25-s + (4.11e6 + 1.37e7i)27-s − 1.69e7i·29-s − 4.33e7·31-s + ⋯ |
| L(s) = 1 | + (−0.813 + 0.581i)3-s − 1.66i·5-s − 0.418·7-s + (0.323 − 0.946i)9-s + 0.0286i·11-s + 0.349·13-s + (0.970 + 1.35i)15-s + 0.229i·17-s − 1.84·19-s + (0.340 − 0.243i)21-s + 1.18i·23-s − 1.78·25-s + (0.286 + 0.958i)27-s − 0.824i·29-s − 1.51·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.7108055240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7108055240\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (197. - 141. i)T \) |
| good | 5 | \( 1 + 5.21e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 + 7.04e3T + 2.82e8T^{2} \) |
| 11 | \( 1 - 4.61e3iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 1.29e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 3.25e5iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 4.56e6T + 6.13e12T^{2} \) |
| 23 | \( 1 - 7.65e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 + 1.69e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 4.33e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 1.05e8T + 4.80e15T^{2} \) |
| 41 | \( 1 + 6.41e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 4.81e7T + 2.16e16T^{2} \) |
| 47 | \( 1 - 3.13e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 1.48e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + 8.65e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 1.43e8T + 7.13e17T^{2} \) |
| 67 | \( 1 - 2.14e9T + 1.82e18T^{2} \) |
| 71 | \( 1 + 7.27e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 1.94e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + 3.31e9T + 9.46e18T^{2} \) |
| 83 | \( 1 + 5.37e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 6.59e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 - 8.93e9T + 7.37e19T^{2} \) |
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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
−10.85931025840839126919747725796, −9.650885269678973729025583434395, −9.053285131202549493895647077874, −7.975248321176261010820176326714, −6.36699759731705259040053966018, −5.54823125470793262973120407868, −4.54460578129692594349906138274, −3.77430981632617362748950769663, −1.74234645925321673767980185300, −0.60077564675472524396816153151,
0.26771647828930184916204446888, 1.92144390484612777130346554501, 2.88943792492982845975910707567, 4.23524366713777311442868271367, 5.82512489193980451899194875471, 6.59721895325357234839354876331, 7.15522294152934733808001792217, 8.425228815623229572442622847543, 10.01308506212935559866401242094, 10.81500846360718613602866406300
