| L(s) = 1 | + (−2.58 + 3.56i)2-s + (−1.49 + 4.58i)3-s + (13.7 + 42.4i)4-s + (−177. + 129. i)5-s + (−12.4 − 17.1i)6-s + (410. − 133. i)7-s + (−454. − 147. i)8-s + (570. + 414. i)9-s − 967. i·10-s + (555. − 1.20e3i)11-s − 215.·12-s + (−731. + 1.00e3i)13-s + (−587. + 1.80e3i)14-s + (−327. − 1.00e3i)15-s + (−606. + 440. i)16-s + (2.61e3 + 3.59e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.323 + 0.445i)2-s + (−0.0551 + 0.169i)3-s + (0.215 + 0.662i)4-s + (−1.42 + 1.03i)5-s + (−0.0577 − 0.0795i)6-s + (1.19 − 0.388i)7-s + (−0.888 − 0.288i)8-s + (0.783 + 0.569i)9-s − 0.967i·10-s + (0.417 − 0.908i)11-s − 0.124·12-s + (−0.332 + 0.458i)13-s + (−0.214 + 0.658i)14-s + (−0.0970 − 0.298i)15-s + (−0.148 + 0.107i)16-s + (0.531 + 0.731i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.903i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.427 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.538338 + 0.850414i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.538338 + 0.850414i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-555. + 1.20e3i)T \) |
| good | 2 | \( 1 + (2.58 - 3.56i)T + (-19.7 - 60.8i)T^{2} \) |
| 3 | \( 1 + (1.49 - 4.58i)T + (-589. - 428. i)T^{2} \) |
| 5 | \( 1 + (177. - 129. i)T + (4.82e3 - 1.48e4i)T^{2} \) |
| 7 | \( 1 + (-410. + 133. i)T + (9.51e4 - 6.91e4i)T^{2} \) |
| 13 | \( 1 + (731. - 1.00e3i)T + (-1.49e6 - 4.59e6i)T^{2} \) |
| 17 | \( 1 + (-2.61e3 - 3.59e3i)T + (-7.45e6 + 2.29e7i)T^{2} \) |
| 19 | \( 1 + (-5.20e3 - 1.69e3i)T + (3.80e7 + 2.76e7i)T^{2} \) |
| 23 | \( 1 + 1.52e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (1.24e4 - 4.03e3i)T + (4.81e8 - 3.49e8i)T^{2} \) |
| 31 | \( 1 + (-2.15e4 - 1.56e4i)T + (2.74e8 + 8.44e8i)T^{2} \) |
| 37 | \( 1 + (1.37e4 + 4.23e4i)T + (-2.07e9 + 1.50e9i)T^{2} \) |
| 41 | \( 1 + (-5.61e4 - 1.82e4i)T + (3.84e9 + 2.79e9i)T^{2} \) |
| 43 | \( 1 + 7.53e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (-3.19e4 + 9.83e4i)T + (-8.72e9 - 6.33e9i)T^{2} \) |
| 53 | \( 1 + (6.54e4 + 4.75e4i)T + (6.84e9 + 2.10e10i)T^{2} \) |
| 59 | \( 1 + (-3.16e4 - 9.73e4i)T + (-3.41e10 + 2.47e10i)T^{2} \) |
| 61 | \( 1 + (1.88e4 + 2.59e4i)T + (-1.59e10 + 4.89e10i)T^{2} \) |
| 67 | \( 1 + 8.14e4T + 9.04e10T^{2} \) |
| 71 | \( 1 + (4.28e4 - 3.11e4i)T + (3.95e10 - 1.21e11i)T^{2} \) |
| 73 | \( 1 + (-4.45e5 + 1.44e5i)T + (1.22e11 - 8.89e10i)T^{2} \) |
| 79 | \( 1 + (2.52e5 - 3.47e5i)T + (-7.51e10 - 2.31e11i)T^{2} \) |
| 83 | \( 1 + (-2.33e5 - 3.21e5i)T + (-1.01e11 + 3.10e11i)T^{2} \) |
| 89 | \( 1 - 5.52e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (1.22e6 + 8.86e5i)T + (2.57e11 + 7.92e11i)T^{2} \) |
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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
−19.40400341338514280652672751282, −18.29485205309380068030979937695, −16.72686847718432002681447824291, −15.62394149160018265211996911631, −14.34403634067867338362282041787, −11.93667310183072828310527128133, −10.88213756226831161024167940792, −8.124941820469964825978285120091, −7.21832924657945474388813628947, −3.83322169876253573730625633018,
1.07678888252086547789628357442, 4.79246053820855641403917117727, 7.69549486677095983777127374767, 9.448164211690635157235756517361, 11.54828632196530207249417433735, 12.27654841125747418355232614971, 14.85128429507858141088572329392, 15.74108603357754352868017918899, 17.71791801443378826256769814414, 18.94932968694992150735816052998
