| L(s) = 1 | + (−4.06 − 12.4i)2-s + (−21.8 − 15.8i)3-s + (−36.1 + 26.2i)4-s + (158. − 487. i)5-s + (−109. + 337. i)6-s + (846. − 615. i)7-s + (−886. − 643. i)8-s + (225. + 693. i)9-s − 6.73e3·10-s + (1.50e3 − 4.15e3i)11-s + 1.20e3·12-s + (2.33e3 + 7.18e3i)13-s + (−1.11e4 − 8.08e3i)14-s + (−1.11e4 + 8.13e3i)15-s + (−6.21e3 + 1.91e4i)16-s + (−4.99e3 + 1.53e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.358 − 1.10i)2-s + (−0.467 − 0.339i)3-s + (−0.282 + 0.205i)4-s + (0.566 − 1.74i)5-s + (−0.207 + 0.637i)6-s + (0.933 − 0.678i)7-s + (−0.611 − 0.444i)8-s + (0.103 + 0.317i)9-s − 2.12·10-s + (0.340 − 0.940i)11-s + 0.201·12-s + (0.294 + 0.907i)13-s + (−1.08 − 0.787i)14-s + (−0.856 + 0.622i)15-s + (−0.379 + 1.16i)16-s + (−0.246 + 0.758i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.285900 + 1.43618i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.285900 + 1.43618i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (21.8 + 15.8i)T \) |
| 11 | \( 1 + (-1.50e3 + 4.15e3i)T \) |
| good | 2 | \( 1 + (4.06 + 12.4i)T + (-103. + 75.2i)T^{2} \) |
| 5 | \( 1 + (-158. + 487. i)T + (-6.32e4 - 4.59e4i)T^{2} \) |
| 7 | \( 1 + (-846. + 615. i)T + (2.54e5 - 7.83e5i)T^{2} \) |
| 13 | \( 1 + (-2.33e3 - 7.18e3i)T + (-5.07e7 + 3.68e7i)T^{2} \) |
| 17 | \( 1 + (4.99e3 - 1.53e4i)T + (-3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (-1.76e4 - 1.28e4i)T + (2.76e8 + 8.50e8i)T^{2} \) |
| 23 | \( 1 - 1.69e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-6.10e4 + 4.43e4i)T + (5.33e9 - 1.64e10i)T^{2} \) |
| 31 | \( 1 + (-2.13e4 - 6.57e4i)T + (-2.22e10 + 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-2.39e5 + 1.74e5i)T + (2.93e10 - 9.02e10i)T^{2} \) |
| 41 | \( 1 + (-5.70e5 - 4.14e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 + 5.81e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-8.94e5 - 6.50e5i)T + (1.56e11 + 4.81e11i)T^{2} \) |
| 53 | \( 1 + (4.71e5 + 1.45e6i)T + (-9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (2.04e6 - 1.48e6i)T + (7.69e11 - 2.36e12i)T^{2} \) |
| 61 | \( 1 + (-2.12e5 + 6.52e5i)T + (-2.54e12 - 1.84e12i)T^{2} \) |
| 67 | \( 1 - 6.20e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (5.11e5 - 1.57e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-1.50e6 + 1.09e6i)T + (3.41e12 - 1.05e13i)T^{2} \) |
| 79 | \( 1 + (-1.03e6 - 3.18e6i)T + (-1.55e13 + 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-2.91e6 + 8.95e6i)T + (-2.19e13 - 1.59e13i)T^{2} \) |
| 89 | \( 1 + 6.19e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (1.03e6 + 3.18e6i)T + (-6.53e13 + 4.74e13i)T^{2} \) |
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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
−14.00639263825881544735048426512, −12.93097445290620041859476655257, −11.85209810795944034904044697745, −10.93249513513625941110849908403, −9.443057485792248397603420656626, −8.322076514359057756783888618084, −6.01016128585342588842812334358, −4.36353987140861139210947434830, −1.58205563984758280110979410535, −0.897296617468419941030051128916,
2.64272468862406980272693005967, 5.35982272975579213335595878380, 6.57368614995731315140534556421, 7.62319898662496142875590457430, 9.395412795328513309373495344768, 10.80300461324076330608290530107, 11.83809460789554539633680317948, 14.06678802815135347629663896997, 15.07817490150607633698641094467, 15.48983634587352537932153478904
