| L(s) = 1 | + (−4 + 6.92i)2-s + (−15 − 25.9i)3-s + (−31.9 − 55.4i)4-s + (62.5 − 108. i)5-s + 240·6-s + (−686 − 594. i)7-s + 511.·8-s + (643.5 − 1.11e3i)9-s + (499. + 866. i)10-s + (−1.5 − 2.59i)11-s + (−960. + 1.66e3i)12-s + 1.74e3·13-s + (6.86e3 − 2.37e3i)14-s − 3.75e3·15-s + (−2.04e3 + 3.54e3i)16-s + (1.89e3 + 3.27e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.320 − 0.555i)3-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + 0.453·6-s + (−0.755 − 0.654i)7-s + 0.353·8-s + (0.294 − 0.509i)9-s + (0.158 + 0.273i)10-s + (−0.000339 − 0.000588i)11-s + (−0.160 + 0.277i)12-s + 0.220·13-s + (0.668 − 0.231i)14-s − 0.286·15-s + (−0.125 + 0.216i)16-s + (0.0934 + 0.161i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0134958 + 0.212666i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0134958 + 0.212666i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4 - 6.92i)T \) |
| 5 | \( 1 + (-62.5 + 108. i)T \) |
| 7 | \( 1 + (686 + 594. i)T \) |
| good | 3 | \( 1 + (15 + 25.9i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 - 1.74e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (-1.89e3 - 3.27e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-972.5 + 1.68e3i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (3.97e4 - 6.88e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + 9.49e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + (6.38e4 + 1.10e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-6.41e4 + 1.11e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 2.98e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.75e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (3.05e5 - 5.29e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-1.29e5 - 2.24e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.43e6 - 2.49e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-7.42e4 + 1.28e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-8.95e5 - 1.55e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 4.93e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-1.02e6 - 1.78e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-2.93e6 + 5.08e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 9.21e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-2.56e6 + 4.43e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 5.87e6T + 8.07e13T^{2} \) |
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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
−12.93977611520657081422370805930, −11.64458232771364880209556318152, −10.11488011783640974444082699597, −9.243915732518817016888872223680, −7.72974003805528706353731811450, −6.70567069635833995357552181114, −5.67848292527352816822849670125, −3.85052014722043819920565316062, −1.39834734024953272993746181855, −0.089610292530517640205285606465,
2.08830017206978964132829642591, 3.55018079325339840572283280200, 5.16633478467232399279743579451, 6.65720519646218543361250358589, 8.319862867057588495377170919900, 9.638503876132355961915939226871, 10.34955384994293440511458567900, 11.42478282048023856762291648759, 12.57841888908371114608834281706, 13.60236286053991764314744351285
