| L(s) = 1 | + (−15.4 + 11.2i)2-s + (72.8 − 224. i)4-s + (−0.205 − 0.149i)5-s + (242. − 746. i)7-s + (635. + 1.95e3i)8-s + 4.84·10-s + (4.41e3 − 157. i)11-s + (3.85e3 − 2.80e3i)13-s + (4.62e3 + 1.42e4i)14-s + (−7.30e3 − 5.31e3i)16-s + (−8.29e3 − 6.02e3i)17-s + (4.57e3 + 1.40e4i)19-s + (−48.4 + 35.2i)20-s + (−6.63e4 + 5.18e4i)22-s − 9.40e4·23-s + ⋯ |
| L(s) = 1 | + (−1.36 + 0.990i)2-s + (0.569 − 1.75i)4-s + (−0.000735 − 0.000534i)5-s + (0.267 − 0.822i)7-s + (0.438 + 1.35i)8-s + 0.00153·10-s + (0.999 − 0.0356i)11-s + (0.486 − 0.353i)13-s + (0.450 + 1.38i)14-s + (−0.446 − 0.324i)16-s + (−0.409 − 0.297i)17-s + (0.152 + 0.470i)19-s + (−0.00135 + 0.000984i)20-s + (−1.32 + 1.03i)22-s − 1.61·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.487740 - 0.349969i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.487740 - 0.349969i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-4.41e3 + 157. i)T \) |
| good | 2 | \( 1 + (15.4 - 11.2i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (0.205 + 0.149i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-242. + 746. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (-3.85e3 + 2.80e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (8.29e3 + 6.02e3i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-4.57e3 - 1.40e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + 9.40e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (3.57e4 - 1.09e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-3.66e4 + 2.66e4i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (6.84e4 - 2.10e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (1.65e5 + 5.10e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 - 5.79e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-1.84e4 - 5.66e4i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-1.37e6 + 9.98e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-2.93e5 + 9.02e5i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (2.12e6 + 1.54e6i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 + 3.39e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-4.89e4 - 3.55e4i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (5.68e5 - 1.74e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (3.05e6 - 2.21e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (4.99e6 + 3.63e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 - 1.26e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-8.22e6 + 5.97e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−12.05952113288172266394214118126, −10.74236207684692920047806475891, −9.929219395826282699948813078139, −8.809615608400793639031354455585, −7.893700910985168270545570796927, −6.88174699668457390388058691762, −5.87744845187703529001403799392, −4.01479926927462651868075532710, −1.54477945768440829938104603100, −0.32299334025486942253697119082,
1.31754158028691218922441557407, 2.37389395663047601489433904177, 3.95216117822505722689501441168, 6.01772354262585728709052335146, 7.59445890593560433282034878098, 8.775972387253124592286279959542, 9.322768508756096147644990755461, 10.52067119864797844094072232184, 11.64463042150692837768759888966, 12.02343179963083265718085953666
