| L(s) = 1 | + (−8.13 − 13.7i)2-s + (61.0 + 25.3i)3-s + (−123. + 224. i)4-s + (−408. + 169. i)5-s + (−148. − 1.04e3i)6-s + (−640. − 640. i)7-s + (4.09e3 − 123. i)8-s + (−1.54e3 − 1.54e3i)9-s + (5.66e3 + 4.25e3i)10-s + (2.58e4 − 1.07e4i)11-s + (−1.32e4 + 1.05e4i)12-s + (2.00e4 + 8.29e3i)13-s + (−3.61e3 + 1.40e4i)14-s − 2.92e4·15-s + (−3.50e4 − 5.53e4i)16-s − 6.73e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.508 − 0.860i)2-s + (0.754 + 0.312i)3-s + (−0.482 + 0.875i)4-s + (−0.654 + 0.270i)5-s + (−0.114 − 0.808i)6-s + (−0.266 − 0.266i)7-s + (0.999 − 0.0301i)8-s + (−0.235 − 0.235i)9-s + (0.566 + 0.425i)10-s + (1.76 − 0.730i)11-s + (−0.637 + 0.509i)12-s + (0.701 + 0.290i)13-s + (−0.0940 + 0.365i)14-s − 0.578·15-s + (−0.534 − 0.845i)16-s − 0.806i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.10934 - 0.987885i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10934 - 0.987885i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (8.13 + 13.7i)T \) |
| good | 3 | \( 1 + (-61.0 - 25.3i)T + (4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (408. - 169. i)T + (2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (640. + 640. i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 + (-2.58e4 + 1.07e4i)T + (1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (-2.00e4 - 8.29e3i)T + (5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 + 6.73e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (-5.65e4 + 1.36e5i)T + (-1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (-2.38e5 + 2.38e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + (1.57e5 - 3.79e5i)T + (-3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 - 7.83e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (-2.04e6 + 8.46e5i)T + (2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (1.20e6 + 1.20e6i)T + 7.98e12iT^{2} \) |
| 43 | \( 1 + (-1.63e6 + 6.75e5i)T + (8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 - 3.35e5T + 2.38e13T^{2} \) |
| 53 | \( 1 + (-2.36e6 - 5.70e6i)T + (-4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (-5.89e6 - 1.42e7i)T + (-1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (-7.92e6 + 1.91e7i)T + (-1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (2.47e7 + 1.02e7i)T + (2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (-1.68e7 - 1.68e7i)T + 6.45e14iT^{2} \) |
| 73 | \( 1 + (6.36e6 + 6.36e6i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 + 5.62e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (1.52e7 - 3.68e7i)T + (-1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (-1.88e7 + 1.88e7i)T - 3.93e15iT^{2} \) |
| 97 | \( 1 - 8.10e7T + 7.83e15T^{2} \) |
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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
−14.52035624161132463898726577688, −13.55408935518649025260086352431, −11.85397009572532399250560746500, −11.07530727864352711816933540780, −9.307422967985736379910139859137, −8.711244614855707077533032715900, −6.97787386897708308574924818823, −3.99389240127459929367773198808, −3.07570692806559378515313359591, −0.806197726583167632497086357844,
1.41580770164214050045586117902, 3.97095611371807354860268952478, 6.05501673435900564248905138534, 7.58316271200667175632889451788, 8.573212965601607854715502555718, 9.652445575101023042717341769331, 11.55265175961248713313938679794, 13.15587475605235924082698384957, 14.43181487747719547821890129921, 15.19480090026984453001970920271
