| L(s) = 1 | + (−130. + 225. i)5-s + (113. + 197. i)7-s + (−1.73e3 − 3.01e3i)11-s + (728. − 1.26e3i)13-s + 1.46e4·17-s − 6.09e3·19-s + (−1.63e3 + 2.83e3i)23-s + (5.03e3 + 8.72e3i)25-s + (5.51e4 + 9.54e4i)29-s + (−1.69e4 + 2.93e4i)31-s − 5.94e4·35-s − 1.05e5·37-s + (−3.37e5 + 5.84e5i)41-s + (3.21e5 + 5.56e5i)43-s + (−1.22e5 − 2.11e5i)47-s + ⋯ |
| L(s) = 1 | + (−0.466 + 0.808i)5-s + (0.125 + 0.217i)7-s + (−0.393 − 0.682i)11-s + (0.0919 − 0.159i)13-s + 0.724·17-s − 0.203·19-s + (−0.0280 + 0.0485i)23-s + (0.0644 + 0.111i)25-s + (0.419 + 0.726i)29-s + (−0.102 + 0.176i)31-s − 0.234·35-s − 0.342·37-s + (−0.764 + 1.32i)41-s + (0.615 + 1.06i)43-s + (−0.171 − 0.297i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1044190350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1044190350\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (130. - 225. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-113. - 197. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (1.73e3 + 3.01e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-728. + 1.26e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 - 1.46e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 6.09e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + (1.63e3 - 2.83e3i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-5.51e4 - 9.54e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (1.69e4 - 2.93e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 1.05e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (3.37e5 - 5.84e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-3.21e5 - 5.56e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (1.22e5 + 2.11e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + 1.83e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.06e6 - 1.84e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-8.33e5 - 1.44e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-2.39e6 + 4.15e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 2.50e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.46e4T + 1.10e13T^{2} \) |
| 79 | \( 1 + (3.33e6 + 5.77e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-9.91e5 - 1.71e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + 8.71e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-4.29e5 - 7.44e5i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−10.66538483693791866159822178098, −9.705248418366727251892780042896, −8.546159302715339526663853201187, −7.80845208957896467666492293702, −6.85638256989256521849206690240, −5.88928190948312246588346087704, −4.84497559255428288626168842545, −3.47573525313644458801105936731, −2.83868756714645475840331805027, −1.35218058683073451135929675661,
0.02300946457484224460840553369, 1.07853013352582536772672680162, 2.29614380281843454317799398715, 3.73024834537724499198593057587, 4.62188006065407989667038544197, 5.48649164883307015923334614866, 6.76302326035627607410677753741, 7.76185226665374827682471763380, 8.439024849532825244295357121321, 9.460354229482977737958516551589
