| L(s) = 1 | + (3.96 + 17.3i)2-s + (−6.73 − 6.25i)3-s + (−170. + 81.9i)4-s + (429. − 64.7i)5-s + (81.7 − 141. i)6-s + (837. + 1.45e3i)7-s + (−675. − 847. i)8-s + (−157. − 2.09e3i)9-s + (2.82e3 + 7.19e3i)10-s + (407. + 196. i)11-s + (1.65e3 + 511. i)12-s + (−2.63e3 + 6.70e3i)13-s + (−2.18e4 + 2.02e4i)14-s + (−3.29e3 − 2.24e3i)15-s + (−3.04e3 + 3.81e3i)16-s + (3.94e3 + 594. i)17-s + ⋯ |
| L(s) = 1 | + (0.350 + 1.53i)2-s + (−0.144 − 0.133i)3-s + (−1.32 + 0.640i)4-s + (1.53 − 0.231i)5-s + (0.154 − 0.267i)6-s + (0.922 + 1.59i)7-s + (−0.466 − 0.585i)8-s + (−0.0718 − 0.958i)9-s + (0.893 + 2.27i)10-s + (0.0924 + 0.0445i)11-s + (0.277 + 0.0854i)12-s + (−0.332 + 0.846i)13-s + (−2.12 + 1.97i)14-s + (−0.252 − 0.171i)15-s + (−0.185 + 0.233i)16-s + (0.194 + 0.0293i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.882268 + 2.53042i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.882268 + 2.53042i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.73e5 + 4.43e5i)T \) |
| good | 2 | \( 1 + (-3.96 - 17.3i)T + (-115. + 55.5i)T^{2} \) |
| 3 | \( 1 + (6.73 + 6.25i)T + (163. + 2.18e3i)T^{2} \) |
| 5 | \( 1 + (-429. + 64.7i)T + (7.46e4 - 2.30e4i)T^{2} \) |
| 7 | \( 1 + (-837. - 1.45e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-407. - 196. i)T + (1.21e7 + 1.52e7i)T^{2} \) |
| 13 | \( 1 + (2.63e3 - 6.70e3i)T + (-4.59e7 - 4.26e7i)T^{2} \) |
| 17 | \( 1 + (-3.94e3 - 594. i)T + (3.92e8 + 1.20e8i)T^{2} \) |
| 19 | \( 1 + (-3.25e3 + 4.34e4i)T + (-8.83e8 - 1.33e8i)T^{2} \) |
| 23 | \( 1 + (2.42e4 - 1.65e4i)T + (1.24e9 - 3.16e9i)T^{2} \) |
| 29 | \( 1 + (1.50e5 - 1.39e5i)T + (1.28e9 - 1.72e10i)T^{2} \) |
| 31 | \( 1 + (-1.34e5 - 4.13e4i)T + (2.27e10 + 1.54e10i)T^{2} \) |
| 37 | \( 1 + (2.58e4 - 4.48e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-9.22e3 - 4.04e4i)T + (-1.75e11 + 8.45e10i)T^{2} \) |
| 47 | \( 1 + (-5.34e5 + 2.57e5i)T + (3.15e11 - 3.96e11i)T^{2} \) |
| 53 | \( 1 + (3.22e5 + 8.21e5i)T + (-8.61e11 + 7.99e11i)T^{2} \) |
| 59 | \( 1 + (-1.68e6 + 2.11e6i)T + (-5.53e11 - 2.42e12i)T^{2} \) |
| 61 | \( 1 + (-2.19e6 + 6.75e5i)T + (2.59e12 - 1.77e12i)T^{2} \) |
| 67 | \( 1 + (2.11e5 - 2.82e6i)T + (-5.99e12 - 9.03e11i)T^{2} \) |
| 71 | \( 1 + (1.88e5 + 1.28e5i)T + (3.32e12 + 8.46e12i)T^{2} \) |
| 73 | \( 1 + (-1.50e6 + 3.82e6i)T + (-8.09e12 - 7.51e12i)T^{2} \) |
| 79 | \( 1 + (1.32e6 + 2.29e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (6.84e6 + 6.35e6i)T + (2.02e12 + 2.70e13i)T^{2} \) |
| 89 | \( 1 + (-1.17e6 - 1.08e6i)T + (3.30e12 + 4.41e13i)T^{2} \) |
| 97 | \( 1 + (-8.07e6 - 3.88e6i)T + (5.03e13 + 6.31e13i)T^{2} \) |
| 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
−14.82716482950053737467296350894, −14.17871387732953715922651364105, −12.94045854887193095600270065443, −11.63093932852214879605867468395, −9.335648240001729584312792016579, −8.727500654049198232053954656778, −6.84624592156410177865218798969, −5.78857729649594020795298656061, −5.03031399271421143984377079184, −2.00856371692417920304936047278,
1.15404483885702719361010057810, 2.29655856098017529304989500144, 4.16684688217094995301186533105, 5.53970105671226736191351793900, 7.73278083441713375436673879377, 10.00990636139623376330400631582, 10.30785417409124650101797493782, 11.30237581543364846135972493287, 12.92366953227177287220292874734, 13.80726075953939689420861188060
