| L(s) = 1 | + (7.18 + 5.22i)2-s + (49.0 − 5.15i)3-s + (4.60 + 14.1i)4-s + (67.5 + 116. i)5-s + (379. + 218. i)6-s + (−500. − 106. i)7-s + (134. − 414. i)8-s + (1.66e3 − 353. i)9-s + (−125. + 1.19e3i)10-s + (−942. − 848. i)11-s + (298. + 670. i)12-s + (−1.17e3 + 2.64e3i)13-s + (−3.04e3 − 3.37e3i)14-s + (3.91e3 + 5.38e3i)15-s + (3.90e3 − 2.83e3i)16-s + (−4.40e3 + 3.96e3i)17-s + ⋯ |
| L(s) = 1 | + (0.898 + 0.652i)2-s + (1.81 − 0.190i)3-s + (0.0719 + 0.221i)4-s + (0.540 + 0.935i)5-s + (1.75 + 1.01i)6-s + (−1.45 − 0.310i)7-s + (0.263 − 0.810i)8-s + (2.27 − 0.484i)9-s + (−0.125 + 1.19i)10-s + (−0.708 − 0.637i)11-s + (0.172 + 0.388i)12-s + (−0.535 + 1.20i)13-s + (−1.10 − 1.23i)14-s + (1.15 + 1.59i)15-s + (0.953 − 0.692i)16-s + (−0.895 + 0.806i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.74100 + 1.23243i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.74100 + 1.23243i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (1.50e4 + 2.57e4i)T \) |
| good | 2 | \( 1 + (-7.18 - 5.22i)T + (19.7 + 60.8i)T^{2} \) |
| 3 | \( 1 + (-49.0 + 5.15i)T + (713. - 151. i)T^{2} \) |
| 5 | \( 1 + (-67.5 - 116. i)T + (-7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (500. + 106. i)T + (1.07e5 + 4.78e4i)T^{2} \) |
| 11 | \( 1 + (942. + 848. i)T + (1.85e5 + 1.76e6i)T^{2} \) |
| 13 | \( 1 + (1.17e3 - 2.64e3i)T + (-3.22e6 - 3.58e6i)T^{2} \) |
| 17 | \( 1 + (4.40e3 - 3.96e3i)T + (2.52e6 - 2.40e7i)T^{2} \) |
| 19 | \( 1 + (-6.75e3 + 3.00e3i)T + (3.14e7 - 3.49e7i)T^{2} \) |
| 23 | \( 1 + (3.80e3 + 1.23e3i)T + (1.19e8 + 8.70e7i)T^{2} \) |
| 29 | \( 1 + (5.28e3 - 7.27e3i)T + (-1.83e8 - 5.65e8i)T^{2} \) |
| 37 | \( 1 + (-3.97e4 - 2.29e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-2.29e3 + 2.18e4i)T + (-4.64e9 - 9.87e8i)T^{2} \) |
| 43 | \( 1 + (1.08e4 + 2.43e4i)T + (-4.22e9 + 4.69e9i)T^{2} \) |
| 47 | \( 1 + (-1.72e4 + 1.25e4i)T + (3.33e9 - 1.02e10i)T^{2} \) |
| 53 | \( 1 + (-3.32e4 - 1.56e5i)T + (-2.02e10 + 9.01e9i)T^{2} \) |
| 59 | \( 1 + (1.71e4 + 1.63e5i)T + (-4.12e10 + 8.76e9i)T^{2} \) |
| 61 | \( 1 - 2.03e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + (-2.58e5 - 4.48e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + (1.44e5 - 3.06e4i)T + (1.17e11 - 5.21e10i)T^{2} \) |
| 73 | \( 1 + (-1.03e5 - 9.32e4i)T + (1.58e10 + 1.50e11i)T^{2} \) |
| 79 | \( 1 + (-2.19e5 + 1.97e5i)T + (2.54e10 - 2.41e11i)T^{2} \) |
| 83 | \( 1 + (-1.55e5 - 1.63e4i)T + (3.19e11 + 6.79e10i)T^{2} \) |
| 89 | \( 1 + (-5.75e5 + 1.86e5i)T + (4.02e11 - 2.92e11i)T^{2} \) |
| 97 | \( 1 + (-1.66e5 - 5.11e5i)T + (-6.73e11 + 4.89e11i)T^{2} \) |
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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
−15.31839731363462005542067056914, −14.33452357366638466874060370093, −13.59844754253246692282603766174, −13.00678659028071984630915037302, −10.17711916364187295409960858388, −9.252965372469213672162254743551, −7.29943447179962214101407073537, −6.40331230889176431704462819253, −3.85496268179960451362914766034, −2.62023701624201057720937662551,
2.36194974552845779808856028107, 3.35098527462563544086284895011, 5.04333238260507610469266594714, 7.74147072022059475192625863755, 9.157689701426455239406048186867, 9.995496895255341579720103730308, 12.61961297393770877443973470066, 13.00419855234589713437179830421, 13.84905340091569802916447450735, 15.21333112909300033683687159320
