| 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
−14.48330138690715871794826322081, −13.27824895850031328587744711330, −12.82714321093783850450424871292, −10.51247539780836412058386555609, −10.02236563682858066495920060666, −8.405667162145819385673874800540, −7.37407874058229086606437361317, −5.31553494745053863685494374616, −3.83302106077292764038199543244, −1.28642543107687012222285020715,
0.43848894426116935979933072800, 2.85257385919899071372435018615, 4.78361175119715785178276174768, 6.41238345233876661346017491209, 8.142788027504784248953828488883, 9.433145858240005789470417881092, 10.07293096744462663205312577102, 11.96291872709527330694410724434, 12.99867855896445161815101823768, 14.21646396415833118604544296821
