| L(s) = 1 | + (10.1 − 21.0i)2-s + (9.28 − 13.6i)3-s + (−179. − 224. i)4-s + (619. + 667. i)5-s + (−192. − 332. i)6-s + (1.11e3 + 641. i)7-s + (−711. + 162. i)8-s + (2.29e3 + 5.85e3i)9-s + (2.02e4 − 6.26e3i)10-s + (−9.04e3 + 1.13e4i)11-s + (−4.72e3 + 354. i)12-s + (3.67e4 + 1.13e4i)13-s + (2.47e4 − 1.68e4i)14-s + (1.48e4 − 2.23e3i)15-s + (1.25e4 − 5.51e4i)16-s + (−1.26e4 − 1.17e4i)17-s + ⋯ |
| L(s) = 1 | + (0.632 − 1.31i)2-s + (0.114 − 0.168i)3-s + (−0.699 − 0.877i)4-s + (0.991 + 1.06i)5-s + (−0.148 − 0.256i)6-s + (0.463 + 0.267i)7-s + (−0.173 + 0.0396i)8-s + (0.350 + 0.892i)9-s + (2.02 − 0.626i)10-s + (−0.617 + 0.774i)11-s + (−0.227 + 0.0170i)12-s + (1.28 + 0.397i)13-s + (0.643 − 0.438i)14-s + (0.293 − 0.0442i)15-s + (0.191 − 0.841i)16-s + (−0.151 − 0.140i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.17749 - 1.34005i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.17749 - 1.34005i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.96e6 - 2.79e6i)T \) |
| good | 2 | \( 1 + (-10.1 + 21.0i)T + (-159. - 200. i)T^{2} \) |
| 3 | \( 1 + (-9.28 + 13.6i)T + (-2.39e3 - 6.10e3i)T^{2} \) |
| 5 | \( 1 + (-619. - 667. i)T + (-2.91e4 + 3.89e5i)T^{2} \) |
| 7 | \( 1 + (-1.11e3 - 641. i)T + (2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (9.04e3 - 1.13e4i)T + (-4.76e7 - 2.08e8i)T^{2} \) |
| 13 | \( 1 + (-3.67e4 - 1.13e4i)T + (6.73e8 + 4.59e8i)T^{2} \) |
| 17 | \( 1 + (1.26e4 + 1.17e4i)T + (5.21e8 + 6.95e9i)T^{2} \) |
| 19 | \( 1 + (-4.96e3 - 1.94e3i)T + (1.24e10 + 1.15e10i)T^{2} \) |
| 23 | \( 1 + (5.03e5 + 7.58e4i)T + (7.48e10 + 2.30e10i)T^{2} \) |
| 29 | \( 1 + (1.49e5 + 2.18e5i)T + (-1.82e11 + 4.65e11i)T^{2} \) |
| 31 | \( 1 + (9.96e4 + 1.32e6i)T + (-8.43e11 + 1.27e11i)T^{2} \) |
| 37 | \( 1 + (-9.86e5 + 5.69e5i)T + (1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-4.18e6 - 2.01e6i)T + (4.97e12 + 6.24e12i)T^{2} \) |
| 47 | \( 1 + (2.11e6 + 2.65e6i)T + (-5.29e12 + 2.32e13i)T^{2} \) |
| 53 | \( 1 + (-8.24e6 + 2.54e6i)T + (5.14e13 - 3.50e13i)T^{2} \) |
| 59 | \( 1 + (1.34e6 - 5.87e6i)T + (-1.32e14 - 6.37e13i)T^{2} \) |
| 61 | \( 1 + (8.59e6 + 6.44e5i)T + (1.89e14 + 2.85e13i)T^{2} \) |
| 67 | \( 1 + (8.89e6 - 2.26e7i)T + (-2.97e14 - 2.76e14i)T^{2} \) |
| 71 | \( 1 + (3.07e6 + 2.03e7i)T + (-6.17e14 + 1.90e14i)T^{2} \) |
| 73 | \( 1 + (-1.02e7 + 3.32e7i)T + (-6.66e14 - 4.54e14i)T^{2} \) |
| 79 | \( 1 + (9.49e6 - 1.64e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (1.77e7 + 1.21e7i)T + (8.22e14 + 2.09e15i)T^{2} \) |
| 89 | \( 1 + (-5.06e7 + 7.43e7i)T + (-1.43e15 - 3.66e15i)T^{2} \) |
| 97 | \( 1 + (-6.39e7 + 8.01e7i)T + (-1.74e15 - 7.64e15i)T^{2} \) |
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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
−13.69815389326349546775246146934, −13.05582439707979105629356289109, −11.53011959553362702039607343816, −10.63477761138432334752921577994, −9.796288784941284122991668542906, −7.70307517162151277165665576780, −5.93542024242683114957907958993, −4.28669718875672325230944394306, −2.47279807941357370402811921922, −1.82309186835498294703328905772,
1.23769161010450200791404254482, 4.00819428324676227599182740231, 5.43501258832974522743973303242, 6.22681880148734766174684486693, 8.005515424781567104464635022973, 9.012525620109921237611806441333, 10.60637030207005370443164904138, 12.59495419148484357872593025474, 13.55057241760959286490831229778, 14.24994697092677076937222121539
