| L(s) = 1 | − 3.30·2-s + (−12.3 + 21.4i)3-s − 21.0·4-s + (−32.1 + 55.6i)5-s + (40.8 − 70.8i)6-s + (−2.83 − 4.90i)7-s + 175.·8-s + (−183. − 318. i)9-s + (106. − 184. i)10-s − 20.0·11-s + (260. − 450. i)12-s + (308. + 534. i)13-s + (9.37 + 16.2i)14-s + (−794. − 1.37e3i)15-s + 92.8·16-s + (−987. − 1.71e3i)17-s + ⋯ |
| L(s) = 1 | − 0.584·2-s + (−0.792 + 1.37i)3-s − 0.657·4-s + (−0.574 + 0.995i)5-s + (0.463 − 0.803i)6-s + (−0.0218 − 0.0378i)7-s + 0.969·8-s + (−0.757 − 1.31i)9-s + (0.336 − 0.582i)10-s − 0.0498·11-s + (0.521 − 0.903i)12-s + (0.505 + 0.876i)13-s + (0.0127 + 0.0221i)14-s + (−0.911 − 1.57i)15-s + 0.0906·16-s + (−0.829 − 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0207 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0207 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0735491 - 0.0750937i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0735491 - 0.0750937i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (8.00e3 + 9.10e3i)T \) |
| good | 2 | \( 1 + 3.30T + 32T^{2} \) |
| 3 | \( 1 + (12.3 - 21.4i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (32.1 - 55.6i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (2.83 + 4.90i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + 20.0T + 1.61e5T^{2} \) |
| 13 | \( 1 + (-308. - 534. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (987. + 1.71e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (488. - 846. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (165. - 286. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-4.08e3 - 7.08e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-2.16e3 + 3.74e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-3.70e3 + 6.42e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.89e4T + 1.15e8T^{2} \) |
| 47 | \( 1 + 7.98e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-1.12e4 + 1.95e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + 2.25e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (-1.05e3 - 1.82e3i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.11e4 - 1.93e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-2.48e3 - 4.30e3i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 + (1.63e4 + 2.83e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-2.63e4 - 4.55e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (5.83e4 - 1.01e5i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-1.93e4 + 3.35e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 7.58e4T + 8.58e9T^{2} \) |
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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
−16.02676910934446679744100229159, −14.91665849235111724010410244615, −13.74702965492259254390468392181, −11.67783464226615999801243762388, −10.81796490663938994219724375303, −9.898078225416571480894647937990, −8.751136621549814788828452014961, −6.89754126995870905153465005004, −4.96195122090520505746854662481, −3.75372048749109356513661220368,
0.090676706930431585445166943626, 1.24135982214275932262921091854, 4.63833533553024383296019160391, 6.26281057406068035469109715874, 7.965358596766216702184794005796, 8.579136707856064003433488037778, 10.49201222566104584962033428328, 11.89912734500939927924946286845, 12.92761005602917417101053622686, 13.45217733045053363112458551893
