| L(s) = 1 | + (251. − 435. i)5-s + (−776.5 − 1.34e3i)7-s + (3.26e3 + 5.65e3i)11-s + (399.5 − 691. i)13-s − 1.15e4·17-s − 3.55e4·19-s + (−5.30e4 + 9.18e4i)23-s + (−8.72e4 − 1.51e5i)25-s + (−7.44e4 − 1.28e5i)29-s + (−3.89e4 + 6.74e4i)31-s − 7.80e5·35-s + 2.37e5·37-s + (−1.89e5 + 3.27e5i)41-s + (1.31e4 + 2.26e4i)43-s + (1.95e5 + 3.38e5i)47-s + ⋯ |
| L(s) = 1 | + (0.899 − 1.55i)5-s + (−0.855 − 1.48i)7-s + (0.740 + 1.28i)11-s + (0.0504 − 0.0873i)13-s − 0.570·17-s − 1.18·19-s + (−0.908 + 1.57i)23-s + (−1.11 − 1.93i)25-s + (−0.566 − 0.981i)29-s + (−0.234 + 0.406i)31-s − 3.07·35-s + 0.770·37-s + (−0.428 + 0.741i)41-s + (0.0251 + 0.0435i)43-s + (0.274 + 0.475i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1315242734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1315242734\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-251. + 435. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (776.5 + 1.34e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-3.26e3 - 5.65e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-399.5 + 691. i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + 1.15e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.55e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (5.30e4 - 9.18e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (7.44e4 + 1.28e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (3.89e4 - 6.74e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 2.37e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (1.89e5 - 3.27e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-1.31e4 - 2.26e4i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-1.95e5 - 3.38e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 - 1.73e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-9.77e4 + 1.69e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (2.11e5 + 3.66e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (9.75e5 - 1.68e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 1.61e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.47e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (2.33e6 + 4.05e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-9.57e5 - 1.65e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 - 2.61e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-4.15e6 - 7.18e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.29539229895570780962688776126, −9.704420779664147451339606101746, −9.072280141625621502413993514434, −7.81294932695099407008732706965, −6.76566935370961925267546585360, −5.82858998345894459654138474409, −4.48584135144476828616858714241, −3.98755934208019445520064829203, −1.99522294159331683082085493065, −1.13170004712335347984809656492,
0.02745322976082743103377431043, 2.10643601620195645948478555221, 2.70594108353856874866530137974, 3.76734082338335979321394098942, 5.72569717626778036259249239178, 6.23226886864861624239414140399, 6.83014640818248770575936564899, 8.573553058953223687418303915021, 9.151786529257745978632345678593, 10.24014285099714035413538715235
