| 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
−18.94932968694992150735816052998, −17.71791801443378826256769814414, −15.74108603357754352868017918899, −14.85128429507858141088572329392, −12.27654841125747418355232614971, −11.54828632196530207249417433735, −9.448164211690635157235756517361, −7.69549486677095983777127374767, −4.79246053820855641403917117727, −1.07678888252086547789628357442,
3.83322169876253573730625633018, 7.21832924657945474388813628947, 8.124941820469964825978285120091, 10.88213756226831161024167940792, 11.93667310183072828310527128133, 14.34403634067867338362282041787, 15.62394149160018265211996911631, 16.72686847718432002681447824291, 18.29485205309380068030979937695, 19.40400341338514280652672751282
