| L(s) = 1 | + (−5.17 − 29.3i)3-s + (80.9 − 67.8i)5-s + (−461. + 798. i)7-s + (1.22e3 − 444. i)9-s + (−3.88e3 − 6.72e3i)11-s + (115. − 656. i)13-s + (−2.41e3 − 2.02e3i)15-s + (−1.72e3 − 628. i)17-s + (−7.28e3 + 2.89e4i)19-s + (2.58e4 + 9.39e3i)21-s + (−4.30e4 − 3.61e4i)23-s + (−1.16e4 + 6.59e4i)25-s + (−5.19e4 − 8.99e4i)27-s + (−1.76e5 + 6.42e4i)29-s + (−5.78e4 + 1.00e5i)31-s + ⋯ |
| L(s) = 1 | + (−0.110 − 0.627i)3-s + (0.289 − 0.242i)5-s + (−0.508 + 0.879i)7-s + (0.558 − 0.203i)9-s + (−0.879 − 1.52i)11-s + (0.0146 − 0.0828i)13-s + (−0.184 − 0.154i)15-s + (−0.0851 − 0.0310i)17-s + (−0.243 + 0.969i)19-s + (0.608 + 0.221i)21-s + (−0.737 − 0.619i)23-s + (−0.148 + 0.844i)25-s + (−0.507 − 0.879i)27-s + (−1.34 + 0.489i)29-s + (−0.348 + 0.604i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0447438 + 0.311558i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0447438 + 0.311558i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (7.28e3 - 2.89e4i)T \) |
| good | 3 | \( 1 + (5.17 + 29.3i)T + (-2.05e3 + 747. i)T^{2} \) |
| 5 | \( 1 + (-80.9 + 67.8i)T + (1.35e4 - 7.69e4i)T^{2} \) |
| 7 | \( 1 + (461. - 798. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (3.88e3 + 6.72e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-115. + 656. i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 17 | \( 1 + (1.72e3 + 628. i)T + (3.14e8 + 2.63e8i)T^{2} \) |
| 23 | \( 1 + (4.30e4 + 3.61e4i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (1.76e5 - 6.42e4i)T + (1.32e10 - 1.10e10i)T^{2} \) |
| 31 | \( 1 + (5.78e4 - 1.00e5i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 - 4.01e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + (9.43e4 + 5.34e5i)T + (-1.83e11 + 6.66e10i)T^{2} \) |
| 43 | \( 1 + (4.32e5 - 3.63e5i)T + (4.72e10 - 2.67e11i)T^{2} \) |
| 47 | \( 1 + (1.19e6 - 4.34e5i)T + (3.88e11 - 3.25e11i)T^{2} \) |
| 53 | \( 1 + (1.02e6 + 8.61e5i)T + (2.03e11 + 1.15e12i)T^{2} \) |
| 59 | \( 1 + (-5.49e5 - 2.00e5i)T + (1.90e12 + 1.59e12i)T^{2} \) |
| 61 | \( 1 + (-1.34e6 - 1.12e6i)T + (5.45e11 + 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-3.87e6 + 1.41e6i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (3.84e6 - 3.22e6i)T + (1.57e12 - 8.95e12i)T^{2} \) |
| 73 | \( 1 + (2.95e5 + 1.67e6i)T + (-1.03e13 + 3.77e12i)T^{2} \) |
| 79 | \( 1 + (1.22e6 + 6.92e6i)T + (-1.80e13 + 6.56e12i)T^{2} \) |
| 83 | \( 1 + (-4.32e6 + 7.49e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-1.83e5 + 1.04e6i)T + (-4.15e13 - 1.51e13i)T^{2} \) |
| 97 | \( 1 + (-1.08e7 - 3.94e6i)T + (6.18e13 + 5.19e13i)T^{2} \) |
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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
−12.73950728461648670932465060617, −11.52409834584721739700002539106, −10.20585414683469981733889530605, −8.948993143454482856192653263900, −7.85498828576168752182352596172, −6.32991766748033704629087006675, −5.46143020571179801729318848429, −3.37163690111783966650389814719, −1.77798664653210940630956114684, −0.10099528206434923855184722892,
2.07180747194701808772215245480, 3.91921718900384595244118176088, 4.99614080791961946982240009871, 6.74260659116908195274478726729, 7.72646514700850890182310548592, 9.744451527287417892456240248417, 10.03010604043828189738210936099, 11.21972437426434285065467270676, 12.82826212682026376832434523215, 13.47177269967282711690167849482
