| L(s) = 1 | + (−225. − 389. i)5-s + (817. − 1.41e3i)7-s + (695. − 1.20e3i)11-s + (−3.17e3 − 5.50e3i)13-s − 3.57e4·17-s − 1.68e4·19-s + (−2.86e4 − 4.96e4i)23-s + (−6.22e4 + 1.07e5i)25-s + (−4.54e4 + 7.86e4i)29-s + (−948. − 1.64e3i)31-s − 7.35e5·35-s + 3.56e4·37-s + (−2.41e5 − 4.18e5i)41-s + (2.87e5 − 4.98e5i)43-s + (−4.25e5 + 7.36e5i)47-s + ⋯ |
| L(s) = 1 | + (−0.805 − 1.39i)5-s + (0.900 − 1.56i)7-s + (0.157 − 0.272i)11-s + (−0.401 − 0.695i)13-s − 1.76·17-s − 0.563·19-s + (−0.490 − 0.850i)23-s + (−0.796 + 1.37i)25-s + (−0.345 + 0.598i)29-s + (−0.00571 − 0.00990i)31-s − 2.90·35-s + 0.115·37-s + (−0.548 − 0.949i)41-s + (0.552 − 0.956i)43-s + (−0.597 + 1.03i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8088966174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8088966174\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (225. + 389. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-817. + 1.41e3i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-695. + 1.20e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (3.17e3 + 5.50e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + 3.57e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.68e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (2.86e4 + 4.96e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (4.54e4 - 7.86e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (948. + 1.64e3i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 3.56e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (2.41e5 + 4.18e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-2.87e5 + 4.98e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (4.25e5 - 7.36e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 - 8.54e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-7.30e5 - 1.26e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-9.08e5 + 1.57e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.22e6 + 2.11e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 3.79e4T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.18e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (-1.99e6 + 3.45e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (1.87e6 - 3.24e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 - 8.13e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-7.20e6 + 1.24e7i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−9.076615256483310075100207667469, −8.351155036303621631201514402532, −7.68247547078279516208517683826, −6.71231872686341116988316232280, −5.13954152463027508933002852302, −4.43266845184926491038262160031, −3.83121307538780540258056211520, −1.95446154086670816594205400136, −0.74666328037627375509966733515, −0.21269718621556445148034204957,
2.00431071370885719190322658436, 2.51353822862482153599924832808, 3.89072453711115859048506483160, 4.86964150628594086494872611432, 6.16276207158305969041988143468, 6.91868291034966595814628625535, 7.928970104006871772764092746876, 8.736640006624006737382822343440, 9.700990106687837506148511244293, 10.94479266854355046993914890416
