| L(s) = 1 | + (−4.89 + 2.82i)2-s + (15.9 − 27.7i)4-s + (150. + 86.9i)5-s + (242 + 419. i)7-s + 181. i·8-s − 984·10-s + (−1.16e3 + 670. i)11-s + (−1.68e3 + 2.91e3i)13-s + (−2.37e3 − 1.36e3i)14-s + (−512. − 886. i)16-s − 12.7i·17-s + 5.74e3·19-s + (4.82e3 − 2.78e3i)20-s + (3.79e3 − 6.56e3i)22-s + (−2.92e3 − 1.68e3i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (1.20 + 0.695i)5-s + (0.705 + 1.22i)7-s + 0.353i·8-s − 0.983·10-s + (−0.872 + 0.503i)11-s + (−0.766 + 1.32i)13-s + (−0.864 − 0.498i)14-s + (−0.125 − 0.216i)16-s − 0.00259i·17-s + 0.837·19-s + (0.602 − 0.347i)20-s + (0.356 − 0.616i)22-s + (−0.240 − 0.138i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.466904599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.466904599\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.89 - 2.82i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-150. - 86.9i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-242 - 419. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (1.16e3 - 670. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (1.68e3 - 2.91e3i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 + 12.7iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 5.74e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (2.92e3 + 1.68e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (2.54e4 - 1.46e4i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-1.98e4 + 3.44e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 - 5.25e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (3.20e4 + 1.85e4i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (1.90e3 + 3.29e3i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (6.65e4 - 3.83e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + 2.38e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (2.16e5 + 1.24e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (6.62e3 + 1.14e4i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (8.44e4 - 1.46e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 5.31e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.36e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-1.75e4 - 3.04e4i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-9.50e3 + 5.48e3i)T + (1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 - 1.29e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-1.60e5 - 2.78e5i)T + (-4.16e11 + 7.21e11i)T^{2} \) |
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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
−11.97317545938944203086751467340, −11.09664938370101564615615232295, −9.813379518028657438605778145927, −9.395420013004829839530426929957, −8.080108359009920319549071538589, −6.95700429007041175589570173908, −5.85694106579166271561873170399, −4.96670137884955378474906263653, −2.48129571842909452547732758644, −1.87019349676116935538279658764,
0.50100569250563293599674935027, 1.51388033223643794669596491723, 2.97279200714756544218034680604, 4.77252660160180811358026313127, 5.76395098654351374966137350914, 7.47130263011980038235758055803, 8.153304922970884920948039414754, 9.500198105288149633544010256217, 10.24450797836067027458956464327, 10.96192080610963839118372792154
