| L(s) = 1 | + (−22.2 − 12.8i)2-s + (201. + 349. i)4-s + (176. − 102. i)5-s + (1.85e3 − 3.20e3i)7-s − 3.78e3i·8-s − 5.24e3·10-s + (−1.39e4 − 8.05e3i)11-s + (1.84e4 + 3.19e4i)13-s + (−8.23e4 + 4.75e4i)14-s + (3.00e3 − 5.21e3i)16-s − 4.75e4i·17-s + 2.13e5·19-s + (7.14e4 + 4.12e4i)20-s + (2.06e5 + 3.58e5i)22-s + (1.43e5 − 8.30e4i)23-s + ⋯ |
| L(s) = 1 | + (−1.39 − 0.802i)2-s + (0.788 + 1.36i)4-s + (0.283 − 0.163i)5-s + (0.771 − 1.33i)7-s − 0.924i·8-s − 0.524·10-s + (−0.952 − 0.550i)11-s + (0.646 + 1.11i)13-s + (−2.14 + 1.23i)14-s + (0.0459 − 0.0795i)16-s − 0.569i·17-s + 1.64·19-s + (0.446 + 0.257i)20-s + (0.883 + 1.52i)22-s + (0.514 − 0.296i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.586361 - 0.920403i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.586361 - 0.920403i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (22.2 + 12.8i)T + (128 + 221. i)T^{2} \) |
| 5 | \( 1 + (-176. + 102. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + (-1.85e3 + 3.20e3i)T + (-2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (1.39e4 + 8.05e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + (-1.84e4 - 3.19e4i)T + (-4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 + 4.75e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 2.13e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + (-1.43e5 + 8.30e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-1.17e6 - 6.80e5i)T + (2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-1.09e5 - 1.89e5i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.15e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (6.43e5 - 3.71e5i)T + (3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-1.46e6 + 2.53e6i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (6.31e6 + 3.64e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 - 2.55e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-4.44e6 + 2.56e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-1.34e6 + 2.33e6i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (3.19e5 + 5.53e5i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 3.72e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.13e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-2.50e7 + 4.33e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-2.83e7 - 1.63e7i)T + (1.12e15 + 1.95e15i)T^{2} \) |
| 89 | \( 1 + 6.85e6iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-4.14e7 + 7.17e7i)T + (-3.91e15 - 6.78e15i)T^{2} \) |
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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
−11.78563397955878064242623508598, −11.02387610416287043491105490494, −10.21566945240792189040549963862, −9.125664671400605952104299927965, −8.043383960647812133313987748414, −7.09437759554388721361674181032, −4.97942566115042481952306905551, −3.19433969798133424180126306520, −1.53695730299472082137329496864, −0.68163941996472602758298268255,
1.07493932029214207006504365762, 2.59034033447728583346044126357, 5.25334732348597192861635616087, 6.18590643500639371989985665871, 7.82256787443287476145522026133, 8.293213582434081099660113804579, 9.564531125979011084497551467060, 10.42638435478901587912888640479, 11.70313507142092281213520345686, 13.05519157143758050331981337786
