| L(s) = 1 | + (−4.75 − 5.96i)2-s + (−3.77 + 4.73i)3-s + (−5.82 + 25.5i)4-s + (−49.8 + 24.0i)5-s + 46.1·6-s + 187.·7-s + (−39.9 + 19.2i)8-s + (45.9 + 201. i)9-s + (380. + 183. i)10-s + (−98.0 − 429. i)11-s + (−98.8 − 123. i)12-s + (551. − 265. i)13-s + (−893. − 1.12e3i)14-s + (74.5 − 326. i)15-s + (1.05e3 + 510. i)16-s + (1.02e3 + 494. i)17-s + ⋯ |
| L(s) = 1 | + (−0.840 − 1.05i)2-s + (−0.242 + 0.303i)3-s + (−0.182 + 0.797i)4-s + (−0.892 + 0.429i)5-s + 0.523·6-s + 1.44·7-s + (−0.220 + 0.106i)8-s + (0.188 + 0.827i)9-s + (1.20 + 0.579i)10-s + (−0.244 − 1.06i)11-s + (−0.198 − 0.248i)12-s + (0.904 − 0.435i)13-s + (−1.21 − 1.52i)14-s + (0.0855 − 0.374i)15-s + (1.03 + 0.498i)16-s + (0.862 + 0.415i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.906843 - 0.245796i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.906843 - 0.245796i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.12e4 + 4.47e3i)T \) |
| good | 2 | \( 1 + (4.75 + 5.96i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (3.77 - 4.73i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (49.8 - 24.0i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 - 187.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (98.0 + 429. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-551. + 265. i)T + (2.31e5 - 2.90e5i)T^{2} \) |
| 17 | \( 1 + (-1.02e3 - 494. i)T + (8.85e5 + 1.11e6i)T^{2} \) |
| 19 | \( 1 + (-459. + 2.01e3i)T + (-2.23e6 - 1.07e6i)T^{2} \) |
| 23 | \( 1 + (-846. - 3.70e3i)T + (-5.79e6 + 2.79e6i)T^{2} \) |
| 29 | \( 1 + (-5.09e3 - 6.39e3i)T + (-4.56e6 + 1.99e7i)T^{2} \) |
| 31 | \( 1 + (-3.39e3 - 4.25e3i)T + (-6.37e6 + 2.79e7i)T^{2} \) |
| 37 | \( 1 - 1.10e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (974. + 1.22e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (-3.17e3 + 1.39e4i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (1.49e4 + 7.22e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (-2.15e4 - 1.03e4i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (1.58e4 - 1.99e4i)T + (-1.87e8 - 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-1.35e3 + 5.92e3i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-5.77e3 + 2.53e4i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (2.40e4 - 1.15e4i)T + (1.29e9 - 1.62e9i)T^{2} \) |
| 79 | \( 1 - 2.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-1.95e3 + 2.45e3i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (-6.73e4 + 8.44e4i)T + (-1.24e9 - 5.44e9i)T^{2} \) |
| 97 | \( 1 + (-2.22e4 - 9.72e4i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
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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
−14.99135364801753604307035544427, −13.56490355537889151297168488230, −11.68891859376865023060226141605, −11.13640285926817104453047973764, −10.47432609415440071151963094835, −8.616263461970048424113135784529, −7.79385561873312847907162032895, −5.25389282334520696889486741612, −3.26499806495442475028250308422, −1.18168202112961928379006355595,
0.936480812269946355072238807735, 4.41128120904797371287161364072, 6.27084883399250574515265430369, 7.74963414152716841203908591921, 8.259951031811348908979195127537, 9.798041399594506412052328796065, 11.65775653908852567934095684826, 12.37443369024262627347587185261, 14.43795415049015493249338965113, 15.26040312318002393368516634751
