| L(s) = 1 | − 10.8·2-s + (5.90 + 10.2i)3-s − 9.73·4-s + (−205. − 355. i)5-s + (−64.2 − 111. i)6-s + (−205. + 355. i)7-s + 1.49e3·8-s + (1.02e3 − 1.77e3i)9-s + (2.23e3 + 3.86e3i)10-s − 6.47e3·11-s + (−57.5 − 99.6i)12-s + (4.64e3 − 8.04e3i)13-s + (2.22e3 − 3.86e3i)14-s + (2.42e3 − 4.20e3i)15-s − 1.50e4·16-s + (−1.56e4 + 2.71e4i)17-s + ⋯ |
| L(s) = 1 | − 0.961·2-s + (0.126 + 0.218i)3-s − 0.0760·4-s + (−0.734 − 1.27i)5-s + (−0.121 − 0.210i)6-s + (−0.225 + 0.391i)7-s + 1.03·8-s + (0.468 − 0.810i)9-s + (0.706 + 1.22i)10-s − 1.46·11-s + (−0.00961 − 0.0166i)12-s + (0.586 − 1.01i)13-s + (0.217 − 0.376i)14-s + (0.185 − 0.321i)15-s − 0.918·16-s + (−0.774 + 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00741 - 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.00741 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.243762 + 0.245577i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.243762 + 0.245577i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.49e5 + 3.86e5i)T \) |
| good | 2 | \( 1 + 10.8T + 128T^{2} \) |
| 3 | \( 1 + (-5.90 - 10.2i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (205. + 355. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (205. - 355. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + 6.47e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-4.64e3 + 8.04e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (1.56e4 - 2.71e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.80e4 - 3.12e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-2.61e4 - 4.53e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (8.87e4 - 1.53e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (9.48e4 + 1.64e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-2.44e5 - 4.24e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + 6.70e4T + 1.94e11T^{2} \) |
| 47 | \( 1 - 7.35e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (7.46e5 + 1.29e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + 1.70e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + (-6.40e5 + 1.10e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.39e6 - 2.41e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-7.20e5 + 1.24e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (1.43e6 - 2.49e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (2.59e6 - 4.48e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-4.35e6 - 7.54e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-1.10e6 - 1.91e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + 1.44e7T + 8.07e13T^{2} \) |
| 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
−15.30282486331617991320256565909, −13.12429551421043604830297851155, −12.61207671495554692126407688896, −10.81318282983310416679870024577, −9.601799942239107044329194961770, −8.534914578589918030832819736907, −7.79571360076725623596134279936, −5.40002455416543498452630462810, −3.86138054196724857228817139753, −1.12134826631346000673670702099,
0.24011923444824549791836237850, 2.53833000030383811122277568377, 4.53267009344572434409739391534, 7.12650314295996529718385890210, 7.61293990238588770224686299014, 9.185661153862083279293681660128, 10.62869226457737650313769012931, 11.13215003946859971172086615847, 13.25778125494619197788299613185, 13.98363933613555319635112407473
