L(s) = 1 | + (14.0 − 24.3i)5-s + (−75.7 − 131. i)7-s + (138. + 240. i)11-s + (−291. + 505. i)13-s + 1.61e3·17-s − 1.36e3·19-s + (−428. + 741. i)23-s + (1.16e3 + 2.02e3i)25-s + (4.26e3 + 7.39e3i)29-s + (1.46e3 − 2.54e3i)31-s − 4.26e3·35-s + 4.03e3·37-s + (9.44e3 − 1.63e4i)41-s + (−1.01e4 − 1.75e4i)43-s + (147. + 256. i)47-s + ⋯ |
L(s) = 1 | + (0.251 − 0.435i)5-s + (−0.583 − 1.01i)7-s + (0.346 + 0.599i)11-s + (−0.479 + 0.829i)13-s + 1.35·17-s − 0.869·19-s + (−0.168 + 0.292i)23-s + (0.373 + 0.646i)25-s + (0.942 + 1.63i)29-s + (0.274 − 0.475i)31-s − 0.587·35-s + 0.484·37-s + (0.877 − 1.52i)41-s + (−0.837 − 1.45i)43-s + (0.00976 + 0.0169i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.471i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.882 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.021598792\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.021598792\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-14.0 + 24.3i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (75.7 + 131. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-138. - 240. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (291. - 505. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 - 1.61e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.36e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (428. - 741. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-4.26e3 - 7.39e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-1.46e3 + 2.54e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 4.03e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-9.44e3 + 1.63e4i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (1.01e4 + 1.75e4i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-147. - 256. i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + 3.03e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-8.61e3 + 1.49e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.28e4 + 2.22e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.31e4 - 2.27e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 7.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.49e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-4.96e4 - 8.59e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (2.50e4 + 4.33e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + 1.36e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-3.33e4 - 5.77e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−10.12003001289081054199037936353, −9.531683870514860052290902055798, −8.542917662065977071172146989374, −7.28705950341043648003341953811, −6.75150863355054329430592952133, −5.44231386640853084306152614779, −4.38574599163132842078585104410, −3.43214368412301148143950989968, −1.86349394982845635563294180919, −0.70499618769518911154069849198,
0.78683999447502801480442605977, 2.50273607851355404842603884389, 3.16200520190747108147789072765, 4.67268451648028196785824038808, 6.01009593986190878796210481384, 6.30115376472958670395335624738, 7.81485985849324569756347367909, 8.536036976538362973416420667349, 9.708000482853899676519509199239, 10.20619045460245526240805162810