| L(s) = 1 | + (−24.3 + 77.2i)3-s + (547. + 301. i)5-s + 1.53e3i·7-s + (−5.37e3 − 3.75e3i)9-s − 7.76e3i·11-s + 2.43e4i·13-s + (−3.66e4 + 3.49e4i)15-s − 8.39e4·17-s − 1.15e5·19-s + (−1.18e5 − 3.73e4i)21-s + 1.86e5·23-s + (2.08e5 + 3.30e5i)25-s + (4.21e5 − 3.24e5i)27-s + 9.07e5i·29-s − 1.08e6·31-s + ⋯ |
| L(s) = 1 | + (−0.300 + 0.953i)3-s + (0.875 + 0.483i)5-s + 0.639i·7-s + (−0.819 − 0.572i)9-s − 0.530i·11-s + 0.852i·13-s + (−0.723 + 0.690i)15-s − 1.00·17-s − 0.884·19-s + (−0.610 − 0.192i)21-s + 0.665·23-s + (0.533 + 0.845i)25-s + (0.792 − 0.610i)27-s + 1.28i·29-s − 1.17·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.6320070950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6320070950\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (24.3 - 77.2i)T \) |
| 5 | \( 1 + (-547. - 301. i)T \) |
| good | 7 | \( 1 - 1.53e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 7.76e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.43e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 8.39e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.15e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 1.86e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 9.07e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.08e6T + 8.52e11T^{2} \) |
| 37 | \( 1 - 5.92e4iT - 3.51e12T^{2} \) |
| 41 | \( 1 - 3.29e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 4.48e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 4.73e5T + 2.38e13T^{2} \) |
| 53 | \( 1 + 1.18e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 3.27e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 2.05e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 3.13e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 9.40e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.64e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 4.17e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 2.83e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 1.11e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 5.09e7iT - 7.83e15T^{2} \) |
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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
−12.46730306769611635665593937379, −11.14148335089233446717722624821, −10.64818957845802724976168297265, −9.262317952587809814178087764995, −8.841185065330107309172215417632, −6.77307628680945095596574131922, −5.85859240089527679192303725712, −4.75766693759900051571097249225, −3.28389712813981384541170910535, −1.97467737155556033485860067052,
0.16608627871058492407984416732, 1.39047954400896949140891242824, 2.52037077116544970956326353418, 4.54869229909500585069498847534, 5.79120865277210205607448915821, 6.77537178070539676814953878025, 7.896557316713802769241891532482, 9.055567262513728979623768596334, 10.32488372828273942983684081620, 11.25894265979981110316976729760
