| L(s) = 1 | + 5.29i·2-s − 12.0·4-s − 11.1i·5-s − 18.5·7-s + 20.8i·8-s + 59.2·10-s − 63.1i·11-s + 61.0·13-s − 98.1i·14-s − 303.·16-s − 272. i·17-s + 61.4·19-s + 134. i·20-s + 334.·22-s − 491. i·23-s + ⋯ |
| L(s) = 1 | + 1.32i·2-s − 0.754·4-s − 0.447i·5-s − 0.377·7-s + 0.325i·8-s + 0.592·10-s − 0.522i·11-s + 0.361·13-s − 0.500i·14-s − 1.18·16-s − 0.942i·17-s + 0.170·19-s + 0.337i·20-s + 0.691·22-s − 0.928i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.756233515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.756233515\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 11.1iT \) |
| 7 | \( 1 + 18.5T \) |
| good | 2 | \( 1 - 5.29iT - 16T^{2} \) |
| 11 | \( 1 + 63.1iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 61.0T + 2.85e4T^{2} \) |
| 17 | \( 1 + 272. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 61.4T + 1.30e5T^{2} \) |
| 23 | \( 1 + 491. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 51.7iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.73e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 944.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.11e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.58e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 666. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.18e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 3.17e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 6.22e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.12e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 70.9iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.55e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 9.52e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 8.89e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.39e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 7.16e3T + 8.85e7T^{2} \) |
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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
−11.17336137062944120832339141681, −9.927622314909952802424243075885, −8.854734147643682649460693355358, −8.216152648476069101247476905631, −7.14517660854974379618817367668, −6.28391655725778274761935798283, −5.40074560801144968398832178968, −4.32850933895355731883502605036, −2.66831685099450726176332127192, −0.62900563776136109087909634511,
1.09187762204574415887521743816, 2.35769107730160476462411146462, 3.40694722454721415249290127301, 4.39075710912122936490716881689, 6.02114545907793477909201176781, 7.01258071323391570951463733063, 8.263552388472502935454401836861, 9.565494589862544729934300267152, 10.07746491384541001406462982760, 11.01370645298065968683617289545
