| L(s) = 1 | + (−6.42 + 11.1i)2-s + (−13.9 + 24.1i)3-s + (−18.4 − 32.0i)4-s − 93.0·5-s + (−178. − 309. i)6-s + (171.5 + 297. i)7-s − 1.16e3·8-s + (706. + 1.22e3i)9-s + (597. − 1.03e3i)10-s + (−1.79e3 + 3.10e3i)11-s + 1.02e3·12-s + (−7.75e3 − 1.62e3i)13-s − 4.40e3·14-s + (1.29e3 − 2.24e3i)15-s + (9.87e3 − 1.71e4i)16-s + (−1.90e4 − 3.30e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.567 + 0.983i)2-s + (−0.297 + 0.515i)3-s + (−0.144 − 0.250i)4-s − 0.332·5-s + (−0.337 − 0.585i)6-s + (0.188 + 0.327i)7-s − 0.807·8-s + (0.322 + 0.559i)9-s + (0.188 − 0.327i)10-s + (−0.405 + 0.702i)11-s + 0.171·12-s + (−0.978 − 0.205i)13-s − 0.429·14-s + (0.0990 − 0.171i)15-s + (0.602 − 1.04i)16-s + (−0.942 − 1.63i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0694892 - 0.0168359i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0694892 - 0.0168359i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-171.5 - 297. i)T \) |
| 13 | \( 1 + (7.75e3 + 1.62e3i)T \) |
| good | 2 | \( 1 + (6.42 - 11.1i)T + (-64 - 110. i)T^{2} \) |
| 3 | \( 1 + (13.9 - 24.1i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + 93.0T + 7.81e4T^{2} \) |
| 11 | \( 1 + (1.79e3 - 3.10e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 17 | \( 1 + (1.90e4 + 3.30e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.69e4 - 2.93e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-1.81e3 + 3.14e3i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-433. + 751. i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 - 1.12e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (4.59e4 - 7.95e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-3.32e5 + 5.75e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-4.42e5 - 7.65e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 - 6.28e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 7.96e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-9.12e5 - 1.58e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (9.00e5 + 1.56e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-3.74e5 + 6.48e5i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (2.44e6 + 4.23e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 4.04e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.52e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.41e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-2.60e6 + 4.50e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (8.25e6 + 1.43e7i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
| 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
−12.41061196721332802052898383545, −11.53573254966478976056606548759, −10.09921208292196541684012823891, −9.237638369811169034314550549369, −7.78921263790531361524202974455, −7.24427443819723126540565727084, −5.61486988343614414022603087361, −4.54309889250721007951000250541, −2.52864771884382701730888630207, −0.03327471247944831392523209282,
1.03497827304847696545460285657, 2.41237156836910592528934762888, 4.00043695116041284586316985676, 5.89879758011457392460774752925, 7.14088349335410748019827974485, 8.509514501535505904046119833205, 9.673098890599199650986681917077, 10.74777230988498543465808418852, 11.57512022397949980671613378762, 12.43427342723198720459717681487
