L(s) = 1 | + 3.16·2-s + (4.74 − 7.64i)3-s − 5.99·4-s + (6.32 + 24.1i)5-s + (15.0 − 24.1i)6-s + 15.2i·7-s − 69.5·8-s + (−36.0 − 72.5i)9-s + (20.0 + 76.4i)10-s + 96.7i·11-s + (−28.4 + 45.8i)12-s − 244. i·13-s + 48.3i·14-s + (214. + 66.3i)15-s − 124.·16-s + 278.·17-s + ⋯ |
L(s) = 1 | + 0.790·2-s + (0.527 − 0.849i)3-s − 0.374·4-s + (0.252 + 0.967i)5-s + (0.416 − 0.671i)6-s + 0.312i·7-s − 1.08·8-s + (−0.444 − 0.895i)9-s + (0.200 + 0.764i)10-s + 0.799i·11-s + (−0.197 + 0.318i)12-s − 1.44i·13-s + 0.246i·14-s + (0.955 + 0.294i)15-s − 0.484·16-s + 0.962·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.294i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.63504 - 0.246577i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.63504 - 0.246577i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.74 + 7.64i)T \) |
| 5 | \( 1 + (-6.32 - 24.1i)T \) |
good | 2 | \( 1 - 3.16T + 16T^{2} \) |
| 7 | \( 1 - 15.2iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 96.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 244. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 278.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 308T + 1.30e5T^{2} \) |
| 23 | \( 1 + 414.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 193. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 32T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.28e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 2.08e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.58e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.44e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 1.41e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 3.96e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 928T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.58e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 4.64e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.22e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 8T + 3.89e7T^{2} \) |
| 83 | \( 1 + 4.43e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 9.28e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 2.50e3iT - 8.85e7T^{2} \) |
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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
−18.27628515125553774618298378779, −17.88377707753321760602638546250, −15.16006712389052846824721981193, −14.37345708194104436932401551287, −13.18488156434099070577097185574, −12.05348935887108114357165880236, −9.766299278577160799377346417051, −7.75234641351409789087364819982, −5.87413455040335138768004992486, −3.07872863286147867877675312894,
3.90366100692623750862261309137, 5.37193848658419878445592533012, 8.579577084164756627338195895756, 9.721460416958051393786941291175, 11.90644124478148003091507148755, 13.63206130947250369010704726116, 14.21576245897445210362723740511, 15.95143369593200129511075512075, 16.95081012828054957580361749229, 18.87618686040980262372545696617