| L(s) = 1 | + (−7.66 − 1.35i)2-s + (72.7 − 35.5i)3-s + (−183. − 66.8i)4-s + (−24.0 − 28.6i)5-s + (−605. + 174. i)6-s + (−2.59e3 + 945. i)7-s + (3.04e3 + 1.75e3i)8-s + (4.02e3 − 5.17e3i)9-s + (145. + 252. i)10-s + (−4.78e3 + 5.70e3i)11-s + (−1.57e4 + 1.66e3i)12-s + (2.68e3 + 1.52e4i)13-s + (2.12e4 − 3.73e3i)14-s + (−2.77e3 − 1.23e3i)15-s + (1.73e4 + 1.45e4i)16-s + (−1.24e5 + 7.20e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.479 − 0.0844i)2-s + (0.898 − 0.439i)3-s + (−0.717 − 0.261i)4-s + (−0.0384 − 0.0458i)5-s + (−0.467 + 0.134i)6-s + (−1.08 + 0.393i)7-s + (0.742 + 0.428i)8-s + (0.614 − 0.789i)9-s + (0.0145 + 0.0252i)10-s + (−0.326 + 0.389i)11-s + (−0.758 + 0.0805i)12-s + (0.0939 + 0.532i)13-s + (0.551 − 0.0973i)14-s + (−0.0547 − 0.0242i)15-s + (0.264 + 0.222i)16-s + (−1.49 + 0.862i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.159444 + 0.305229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.159444 + 0.305229i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-72.7 + 35.5i)T \) |
| good | 2 | \( 1 + (7.66 + 1.35i)T + (240. + 87.5i)T^{2} \) |
| 5 | \( 1 + (24.0 + 28.6i)T + (-6.78e4 + 3.84e5i)T^{2} \) |
| 7 | \( 1 + (2.59e3 - 945. i)T + (4.41e6 - 3.70e6i)T^{2} \) |
| 11 | \( 1 + (4.78e3 - 5.70e3i)T + (-3.72e7 - 2.11e8i)T^{2} \) |
| 13 | \( 1 + (-2.68e3 - 1.52e4i)T + (-7.66e8 + 2.78e8i)T^{2} \) |
| 17 | \( 1 + (1.24e5 - 7.20e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (1.01e5 - 1.75e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (3.96e4 - 1.09e5i)T + (-5.99e10 - 5.03e10i)T^{2} \) |
| 29 | \( 1 + (6.09e5 + 1.07e5i)T + (4.70e11 + 1.71e11i)T^{2} \) |
| 31 | \( 1 + (-1.01e6 - 3.69e5i)T + (6.53e11 + 5.48e11i)T^{2} \) |
| 37 | \( 1 + (9.78e5 + 1.69e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (1.39e6 - 2.45e5i)T + (7.50e12 - 2.73e12i)T^{2} \) |
| 43 | \( 1 + (-2.04e6 - 1.71e6i)T + (2.02e12 + 1.15e13i)T^{2} \) |
| 47 | \( 1 + (1.82e6 + 5.00e6i)T + (-1.82e13 + 1.53e13i)T^{2} \) |
| 53 | \( 1 - 3.45e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (1.16e7 + 1.38e7i)T + (-2.54e13 + 1.44e14i)T^{2} \) |
| 61 | \( 1 + (-1.16e7 + 4.24e6i)T + (1.46e14 - 1.23e14i)T^{2} \) |
| 67 | \( 1 + (1.92e6 + 1.09e7i)T + (-3.81e14 + 1.38e14i)T^{2} \) |
| 71 | \( 1 + (2.34e7 - 1.35e7i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (1.28e7 - 2.22e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (9.81e5 - 5.56e6i)T + (-1.42e15 - 5.18e14i)T^{2} \) |
| 83 | \( 1 + (-5.90e6 - 1.04e6i)T + (2.11e15 + 7.70e14i)T^{2} \) |
| 89 | \( 1 + (-3.29e7 - 1.90e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-9.92e6 - 8.32e6i)T + (1.36e15 + 7.71e15i)T^{2} \) |
| 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
−15.83267093970974683670725797658, −14.59575410310116449876523717835, −13.37901966267366864528259930386, −12.52172737440673978497787266215, −10.28088257039123178392234677660, −9.193181327272209066759563367484, −8.210835895826592811566869909030, −6.40940342677719837275976757250, −4.03630262981220749329517349876, −1.97710710095638405735819149645,
0.16079321547301351116003851066, 3.00934493171064538717909830079, 4.52290448354972677157800188335, 7.10282207279044634081225995468, 8.607352481694495891703357058843, 9.487241113398487381278450505604, 10.70310914699104710005836531873, 13.20556825442386132605037310224, 13.46446633023929223123398211571, 15.26940855979527122687613140787
