| L(s) = 1 | − 9.69·2-s + (−18.8 + 32.6i)3-s − 33.9·4-s + (119. − 207. i)5-s + (182. − 316. i)6-s + (493. + 854. i)7-s + 1.57e3·8-s + (383. + 664. i)9-s + (−1.16e3 + 2.00e3i)10-s + 71.3·11-s + (640. − 1.10e3i)12-s + (−1.63e3 − 2.83e3i)13-s + (−4.78e3 − 8.29e3i)14-s + (4.50e3 + 7.81e3i)15-s − 1.08e4·16-s + (−1.15e4 − 2.00e4i)17-s + ⋯ |
| L(s) = 1 | − 0.857·2-s + (−0.402 + 0.697i)3-s − 0.265·4-s + (0.428 − 0.741i)5-s + (0.345 − 0.598i)6-s + (0.543 + 0.942i)7-s + 1.08·8-s + (0.175 + 0.303i)9-s + (−0.366 + 0.635i)10-s + 0.0161·11-s + (0.106 − 0.185i)12-s + (−0.206 − 0.357i)13-s + (−0.466 − 0.807i)14-s + (0.344 + 0.597i)15-s − 0.664·16-s + (−0.571 − 0.989i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0630913 + 0.437175i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0630913 + 0.437175i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (4.68e5 - 2.29e5i)T \) |
| good | 2 | \( 1 + 9.69T + 128T^{2} \) |
| 3 | \( 1 + (18.8 - 32.6i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-119. + 207. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-493. - 854. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 - 71.3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (1.63e3 + 2.83e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (1.15e4 + 2.00e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.44e4 - 4.24e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (1.13e4 - 1.96e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (4.61e4 + 7.99e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (4.83e4 - 8.36e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (3.54e4 - 6.13e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 - 1.83e5T + 1.94e11T^{2} \) |
| 47 | \( 1 - 3.11e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (7.90e5 - 1.36e6i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + 1.46e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + (1.66e6 + 2.87e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.51e6 - 2.62e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.57e5 - 2.72e5i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (9.73e5 + 1.68e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-1.65e6 - 2.86e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (1.82e6 - 3.15e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (3.32e6 - 5.76e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 6.68e6T + 8.07e13T^{2} \) |
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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
−15.27812712729099727433911313715, −13.79461780155720627176237122466, −12.53260647495948140310446514831, −11.06041442406096990819797737190, −9.887745826386216639455338022806, −9.003940453885263810249395041513, −7.894442979777837092987069394586, −5.51557275789624000123212512188, −4.56015097311860029173970362747, −1.70904209127068864596160082205,
0.26852098293748115451711460431, 1.73994838425416104930136144479, 4.35684683097276755126743893191, 6.52399187705781487092016389615, 7.45019708703843689113733770124, 8.921286289908894590305700161171, 10.33239346918909741817116204486, 11.12138240009678890367079777424, 12.86560195021677576342590497505, 13.80398130698818782450190571514
