| L(s) = 1 | + (−5.10 − 10.5i)2-s + (−0.905 − 1.32i)3-s + (−46.2 + 58.0i)4-s + (140. − 151. i)5-s + (−9.44 + 16.3i)6-s + (243. − 140. i)7-s + (116. + 26.6i)8-s + (265. − 676. i)9-s + (−2.32e3 − 716. i)10-s + (−66.0 − 82.7i)11-s + (118. + 8.90i)12-s + (1.51e3 − 468. i)13-s + (−2.72e3 − 1.86e3i)14-s + (−328. − 49.4i)15-s + (743. + 3.25e3i)16-s + (−6.42e3 + 5.96e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.637 − 1.32i)2-s + (−0.0335 − 0.0491i)3-s + (−0.722 + 0.906i)4-s + (1.12 − 1.21i)5-s + (−0.0437 + 0.0757i)6-s + (0.709 − 0.409i)7-s + (0.228 + 0.0520i)8-s + (0.364 − 0.927i)9-s + (−2.32 − 0.716i)10-s + (−0.0495 − 0.0621i)11-s + (0.0687 + 0.00515i)12-s + (0.691 − 0.213i)13-s + (−0.994 − 0.677i)14-s + (−0.0973 − 0.0146i)15-s + (0.181 + 0.794i)16-s + (−1.30 + 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0435i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.999 - 0.0435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0311438 + 1.42890i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0311438 + 1.42890i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-7.89e4 + 9.25e3i)T \) |
| good | 2 | \( 1 + (5.10 + 10.5i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (0.905 + 1.32i)T + (-266. + 678. i)T^{2} \) |
| 5 | \( 1 + (-140. + 151. i)T + (-1.16e3 - 1.55e4i)T^{2} \) |
| 7 | \( 1 + (-243. + 140. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (66.0 + 82.7i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (-1.51e3 + 468. i)T + (3.98e6 - 2.71e6i)T^{2} \) |
| 17 | \( 1 + (6.42e3 - 5.96e3i)T + (1.80e6 - 2.40e7i)T^{2} \) |
| 19 | \( 1 + (-3.77e3 + 1.47e3i)T + (3.44e7 - 3.19e7i)T^{2} \) |
| 23 | \( 1 + (9.63e3 - 1.45e3i)T + (1.41e8 - 4.36e7i)T^{2} \) |
| 29 | \( 1 + (8.90e3 - 1.30e4i)T + (-2.17e8 - 5.53e8i)T^{2} \) |
| 31 | \( 1 + (-2.82e3 + 3.76e4i)T + (-8.77e8 - 1.32e8i)T^{2} \) |
| 37 | \( 1 + (-3.60e4 - 2.08e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (9.66e4 - 4.65e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-6.59e4 + 8.27e4i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-1.36e5 - 4.20e4i)T + (1.83e10 + 1.24e10i)T^{2} \) |
| 59 | \( 1 + (-5.41e4 - 2.37e5i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-1.38e5 + 1.04e4i)T + (5.09e10 - 7.67e9i)T^{2} \) |
| 67 | \( 1 + (1.70e5 + 4.34e5i)T + (-6.63e10 + 6.15e10i)T^{2} \) |
| 71 | \( 1 + (2.15e4 - 1.42e5i)T + (-1.22e11 - 3.77e10i)T^{2} \) |
| 73 | \( 1 + (-1.18e5 - 3.84e5i)T + (-1.25e11 + 8.52e10i)T^{2} \) |
| 79 | \( 1 + (-1.49e5 - 2.59e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-4.93e5 + 3.36e5i)T + (1.19e11 - 3.04e11i)T^{2} \) |
| 89 | \( 1 + (2.94e5 + 4.32e5i)T + (-1.81e11 + 4.62e11i)T^{2} \) |
| 97 | \( 1 + (-7.63e4 - 9.56e4i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−13.50664273806379055159809859078, −12.79690901958811528545900076506, −11.59434385138949059853272749105, −10.37012557516340845350279135468, −9.324954144472153087197261226247, −8.455653146881438728069284339401, −6.02407941833002969433192092480, −4.12787764609188315605896095188, −1.85928435923330141492342267716, −0.883176954409659073392192551057,
2.25099523410943317952732347510, 5.25997054556890949238607415608, 6.47184997704333722370902664228, 7.52182878416354706483212345948, 8.927952334345928966969586666459, 10.16282137703460313795406109403, 11.36875288701399781783191120756, 13.65382038612479739515980639924, 14.25913180674855201682001878688, 15.45693853469292030223389091866
