| L(s) = 1 | + (43.9 − 15.9i)3-s + (130. − 225. i)5-s + (−113. − 197. i)7-s + (1.67e3 − 1.40e3i)9-s + (−1.73e3 − 3.01e3i)11-s + (728. − 1.26e3i)13-s + (2.13e3 − 1.20e4i)15-s − 1.46e4·17-s + 6.09e3·19-s + (−8.15e3 − 6.86e3i)21-s + (−1.63e3 + 2.83e3i)23-s + (5.03e3 + 8.72e3i)25-s + (5.14e4 − 8.84e4i)27-s + (−5.51e4 − 9.54e4i)29-s + (1.69e4 − 2.93e4i)31-s + ⋯ |
| L(s) = 1 | + (0.940 − 0.341i)3-s + (0.466 − 0.808i)5-s + (−0.125 − 0.217i)7-s + (0.767 − 0.641i)9-s + (−0.393 − 0.682i)11-s + (0.0919 − 0.159i)13-s + (0.163 − 0.918i)15-s − 0.724·17-s + 0.203·19-s + (−0.192 − 0.161i)21-s + (−0.0280 + 0.0485i)23-s + (0.0644 + 0.111i)25-s + (0.502 − 0.864i)27-s + (−0.419 − 0.726i)29-s + (0.102 − 0.176i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.66028 - 1.98288i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66028 - 1.98288i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-43.9 + 15.9i)T \) |
| 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 | \( 1 + (-4.29e5 - 7.44e5i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
| 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
−13.33322433057654524076802767092, −12.03382744316789052717588348281, −10.43913140203606568058648343336, −9.190489352846134704350505827917, −8.426655663357742089509278015697, −7.11780785830671137415969811211, −5.55527057981047728064253663390, −3.88936142087890768928894422513, −2.30066689810280400919025096638, −0.793299496064183724279744059990,
2.00124211569920489911706671044, 3.11815778662942265690094067780, 4.71518408868993297414418972235, 6.50883209440623956780577501810, 7.70984012789235399209706434578, 9.065357163459289928214693064452, 10.01826987314991330909378257801, 10.98986749288381980673870967866, 12.64977683922760250476163081810, 13.66845090911873537806532590747
