| L(s) = 1 | − 4.76·2-s + (2.16 + 3.74i)3-s − 105.·4-s + (48.3 + 83.8i)5-s + (−10.3 − 17.8i)6-s + (−545. + 944. i)7-s + 1.11e3·8-s + (1.08e3 − 1.87e3i)9-s + (−230. − 399. i)10-s − 3.38e3·11-s + (−227. − 393. i)12-s + (−680. + 1.17e3i)13-s + (2.59e3 − 4.50e3i)14-s + (−209. + 362. i)15-s + 8.17e3·16-s + (1.67e4 − 2.89e4i)17-s + ⋯ |
| L(s) = 1 | − 0.421·2-s + (0.0461 + 0.0800i)3-s − 0.822·4-s + (0.173 + 0.299i)5-s + (−0.0194 − 0.0337i)6-s + (−0.600 + 1.04i)7-s + 0.767·8-s + (0.495 − 0.858i)9-s + (−0.0729 − 0.126i)10-s − 0.767·11-s + (−0.0379 − 0.0658i)12-s + (−0.0858 + 0.148i)13-s + (0.253 − 0.438i)14-s + (−0.0159 + 0.0277i)15-s + 0.498·16-s + (0.826 − 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.626201 - 0.472740i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.626201 - 0.472740i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (2.33e5 - 4.66e5i)T \) |
| good | 2 | \( 1 + 4.76T + 128T^{2} \) |
| 3 | \( 1 + (-2.16 - 3.74i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-48.3 - 83.8i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (545. - 944. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + 3.38e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (680. - 1.17e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-1.67e4 + 2.89e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (7.37e3 + 1.27e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (2.55e4 + 4.42e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-1.10e5 + 1.91e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-1.84e4 - 3.20e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-1.60e5 - 2.77e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 - 3.92e4T + 1.94e11T^{2} \) |
| 47 | \( 1 + 9.99e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-1.15e5 - 2.00e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 - 1.32e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + (-1.18e6 + 2.05e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (2.20e6 + 3.81e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.26e6 + 2.19e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (6.76e5 - 1.17e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-8.37e5 + 1.45e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (6.72e5 + 1.16e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-2.34e6 - 4.06e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + 1.69e7T + 8.07e13T^{2} \) |
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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
−14.21646396415833118604544296821, −12.99867855896445161815101823768, −11.96291872709527330694410724434, −10.07293096744462663205312577102, −9.433145858240005789470417881092, −8.142788027504784248953828488883, −6.41238345233876661346017491209, −4.78361175119715785178276174768, −2.85257385919899071372435018615, −0.43848894426116935979933072800,
1.28642543107687012222285020715, 3.83302106077292764038199543244, 5.31553494745053863685494374616, 7.37407874058229086606437361317, 8.405667162145819385673874800540, 10.02236563682858066495920060666, 10.51247539780836412058386555609, 12.82714321093783850450424871292, 13.27824895850031328587744711330, 14.48330138690715871794826322081
