| L(s) = 1 | + (7.11 − 12.3i)2-s + (45.7 + 9.84i)3-s + (−37.2 − 64.5i)4-s + (−145. − 251. i)5-s + (446. − 493. i)6-s + (−555. + 962. i)7-s + 760.·8-s + (1.99e3 + 900. i)9-s − 4.13e3·10-s + (−2.24e3 + 3.88e3i)11-s + (−1.06e3 − 3.31e3i)12-s + (−1.21e3 − 2.11e3i)13-s + (7.91e3 + 1.37e4i)14-s + (−4.16e3 − 1.29e4i)15-s + (1.01e4 − 1.76e4i)16-s + 1.59e4·17-s + ⋯ |
| L(s) = 1 | + (0.628 − 1.08i)2-s + (0.977 + 0.210i)3-s + (−0.291 − 0.504i)4-s + (−0.519 − 0.900i)5-s + (0.844 − 0.932i)6-s + (−0.612 + 1.06i)7-s + 0.525·8-s + (0.911 + 0.411i)9-s − 1.30·10-s + (−0.508 + 0.880i)11-s + (−0.178 − 0.554i)12-s + (−0.153 − 0.266i)13-s + (0.770 + 1.33i)14-s + (−0.318 − 0.989i)15-s + (0.621 − 1.07i)16-s + 0.785·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.82427 - 1.14675i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.82427 - 1.14675i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-45.7 - 9.84i)T \) |
| good | 2 | \( 1 + (-7.11 + 12.3i)T + (-64 - 110. i)T^{2} \) |
| 5 | \( 1 + (145. + 251. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (555. - 962. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (2.24e3 - 3.88e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (1.21e3 + 2.11e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 - 1.59e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.99e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (3.46e4 + 6.00e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-4.70e4 + 8.14e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-9.96e3 - 1.72e4i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 3.31e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (1.21e5 + 2.09e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (4.15e5 - 7.20e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-8.00e4 + 1.38e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 - 3.11e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-1.56e5 - 2.70e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.87e4 + 4.97e4i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-2.05e6 - 3.55e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 4.03e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 8.23e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + (4.89e5 - 8.47e5i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-1.85e6 + 3.20e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 - 2.09e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (1.75e6 - 3.03e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−19.90104823983494680296731421903, −18.87292620970793225420762467189, −16.26957314914959050142803339280, −14.89788919144736616683792895215, −12.92381657039604754444817662014, −12.32568613250580131213756304168, −10.03617490511455521901862043913, −8.276013555757405353068412327271, −4.43771397850938260922690859735, −2.47325912811518858821550330101,
3.68594789234074481203562033969, 6.68614871522585673470998875876, 7.86420492395915309290052037863, 10.46591462628459761217767585667, 13.24697888364272505059802943890, 14.26174146841049449524055371812, 15.34328996635069367174927316727, 16.63696252539739149084823006415, 18.88247433354971903164448439295, 19.78359164882507744720197800578
