| L(s) = 1 | + (−5.37 − 6.73i)2-s + (15.6 − 19.5i)3-s + (−9.39 + 41.1i)4-s + (−42.9 + 20.6i)5-s − 215.·6-s − 55.8·7-s + (79.3 − 38.1i)8-s + (−85.6 − 375. i)9-s + (369. + 177. i)10-s + (−41.5 − 182. i)11-s + (659. + 827. i)12-s + (−716. + 344. i)13-s + (299. + 376. i)14-s + (−265. + 1.16e3i)15-s + (533. + 257. i)16-s + (820. + 394. i)17-s + ⋯ |
| L(s) = 1 | + (−0.949 − 1.19i)2-s + (1.00 − 1.25i)3-s + (−0.293 + 1.28i)4-s + (−0.767 + 0.369i)5-s − 2.44·6-s − 0.430·7-s + (0.438 − 0.211i)8-s + (−0.352 − 1.54i)9-s + (1.16 + 0.562i)10-s + (−0.103 − 0.454i)11-s + (1.32 + 1.65i)12-s + (−1.17 + 0.566i)13-s + (0.409 + 0.512i)14-s + (−0.304 + 1.33i)15-s + (0.521 + 0.251i)16-s + (0.688 + 0.331i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.312846 + 0.435974i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.312846 + 0.435974i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-5.57e3 - 1.07e4i)T \) |
| good | 2 | \( 1 + (5.37 + 6.73i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (-15.6 + 19.5i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (42.9 - 20.6i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 + 55.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + (41.5 + 182. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (716. - 344. i)T + (2.31e5 - 2.90e5i)T^{2} \) |
| 17 | \( 1 + (-820. - 394. i)T + (8.85e5 + 1.11e6i)T^{2} \) |
| 19 | \( 1 + (-255. + 1.11e3i)T + (-2.23e6 - 1.07e6i)T^{2} \) |
| 23 | \( 1 + (1.08e3 + 4.77e3i)T + (-5.79e6 + 2.79e6i)T^{2} \) |
| 29 | \( 1 + (623. + 782. i)T + (-4.56e6 + 1.99e7i)T^{2} \) |
| 31 | \( 1 + (-854. - 1.07e3i)T + (-6.37e6 + 2.79e7i)T^{2} \) |
| 37 | \( 1 + 1.41e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (1.02e4 + 1.28e4i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (-4.58e3 + 2.01e4i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-1.81e4 - 8.71e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (-4.50e3 - 2.17e3i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (2.17e4 - 2.73e4i)T + (-1.87e8 - 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-1.51e4 + 6.65e4i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (3.10e3 - 1.35e4i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (-4.58e4 + 2.20e4i)T + (1.29e9 - 1.62e9i)T^{2} \) |
| 79 | \( 1 - 4.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-2.76e4 + 3.47e4i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (2.01e4 - 2.52e4i)T + (-1.24e9 - 5.44e9i)T^{2} \) |
| 97 | \( 1 + (2.10e4 + 9.22e4i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
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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
−13.90212851607453621094942701904, −12.49959500503104910528691955625, −11.92435728195949557980360272417, −10.42594537019777529867686110554, −9.034626105609373178109254456071, −8.055473311616410486268656973534, −6.93017970776983003584402555942, −3.32161228023824499488928080048, −2.16517192286703372078652140492, −0.33855231723819568864670057986,
3.48729370649887662904176238615, 5.22902756885025302207501475347, 7.46117604809850996304589457742, 8.240997667632548733020585987466, 9.552412279673045477559678192762, 10.02870620550899682286238644409, 12.19413533991042520876682873312, 14.14018392123098067111564795526, 15.18374948883173564197377050935, 15.69778512418818718119532465416
