| L(s) = 1 | + (3.53 + 7.33i)2-s + (−9.42 + 19.5i)3-s + (−1.45 + 1.82i)4-s + (197. + 45.1i)5-s − 176.·6-s + 222. i·7-s + (489. + 111. i)8-s + (160. + 200. i)9-s + (367. + 1.60e3i)10-s + (−1.44e3 − 1.81e3i)11-s + (−21.9 − 45.6i)12-s + (−628. + 2.75e3i)13-s + (−1.63e3 + 785. i)14-s + (−2.74e3 + 3.44e3i)15-s + (943. + 4.13e3i)16-s + (−1.36e3 − 5.97e3i)17-s + ⋯ |
| L(s) = 1 | + (0.441 + 0.917i)2-s + (−0.349 + 0.725i)3-s + (−0.0227 + 0.0284i)4-s + (1.58 + 0.360i)5-s − 0.819·6-s + 0.648i·7-s + (0.956 + 0.218i)8-s + (0.219 + 0.275i)9-s + (0.367 + 1.60i)10-s + (−1.08 − 1.36i)11-s + (−0.0127 − 0.0264i)12-s + (−0.286 + 1.25i)13-s + (−0.594 + 0.286i)14-s + (−0.813 + 1.02i)15-s + (0.230 + 1.00i)16-s + (−0.277 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.24659 + 2.28498i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24659 + 2.28498i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (7.08e4 + 3.60e4i)T \) |
| good | 2 | \( 1 + (-3.53 - 7.33i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (9.42 - 19.5i)T + (-454. - 569. i)T^{2} \) |
| 5 | \( 1 + (-197. - 45.1i)T + (1.40e4 + 6.77e3i)T^{2} \) |
| 7 | \( 1 - 222. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + (1.44e3 + 1.81e3i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (628. - 2.75e3i)T + (-4.34e6 - 2.09e6i)T^{2} \) |
| 17 | \( 1 + (1.36e3 + 5.97e3i)T + (-2.17e7 + 1.04e7i)T^{2} \) |
| 19 | \( 1 + (3.37e3 + 2.69e3i)T + (1.04e7 + 4.58e7i)T^{2} \) |
| 23 | \( 1 + (-6.83e3 - 8.57e3i)T + (-3.29e7 + 1.44e8i)T^{2} \) |
| 29 | \( 1 + (-1.04e4 - 2.16e4i)T + (-3.70e8 + 4.65e8i)T^{2} \) |
| 31 | \( 1 + (-2.06e4 + 9.95e3i)T + (5.53e8 - 6.93e8i)T^{2} \) |
| 37 | \( 1 + 5.91e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-7.41e3 + 3.57e3i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-1.12e5 + 1.41e5i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (3.96e4 + 1.73e5i)T + (-1.99e10 + 9.61e9i)T^{2} \) |
| 59 | \( 1 + (-3.26e3 - 1.42e4i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (1.53e5 - 3.18e5i)T + (-3.21e10 - 4.02e10i)T^{2} \) |
| 67 | \( 1 + (-3.94e4 + 4.95e4i)T + (-2.01e10 - 8.81e10i)T^{2} \) |
| 71 | \( 1 + (3.19e5 + 2.54e5i)T + (2.85e10 + 1.24e11i)T^{2} \) |
| 73 | \( 1 + (-4.97e5 - 1.13e5i)T + (1.36e11 + 6.56e10i)T^{2} \) |
| 79 | \( 1 + 2.44e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + (-5.53e5 - 2.66e5i)T + (2.03e11 + 2.55e11i)T^{2} \) |
| 89 | \( 1 + (2.25e5 - 4.68e5i)T + (-3.09e11 - 3.88e11i)T^{2} \) |
| 97 | \( 1 + (5.15e4 + 6.45e4i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−15.21635911000679158895426177202, −13.88232990097712495889554157488, −13.46562939682952799053816581592, −11.19748442156156973713786795120, −10.27595137600154986986739397502, −9.023058024475835971349485234447, −6.97030433031712453250664376438, −5.70432305018181059610351100269, −5.03024669462116231419541635762, −2.28363931965786719638823952229,
1.25727618267761136571924130320, 2.44060610083032857560999578003, 4.69799168766071468053147265359, 6.32692720409795898124461095628, 7.76385152968472656213681065522, 10.04249176899055463095273989667, 10.48625828937526622801252592678, 12.52579076257122116321697990397, 12.79000851829243686738230972818, 13.61881409525611828416735344275
