| L(s) = 1 | + (−10.4 − 7.58i)3-s + (−8.80 + 27.1i)5-s + (7.94 + 10.9i)7-s + (26.4 + 81.3i)9-s + (73.2 − 96.3i)11-s + (75.1 − 24.4i)13-s + (297. − 216. i)15-s + (56.6 + 18.3i)17-s + (290. − 399. i)19-s − 174. i·21-s + 496.·23-s + (−151. − 109. i)25-s + (17.9 − 55.3i)27-s + (356. + 490. i)29-s + (508. + 1.56e3i)31-s + ⋯ |
| L(s) = 1 | + (−1.15 − 0.842i)3-s + (−0.352 + 1.08i)5-s + (0.162 + 0.223i)7-s + (0.326 + 1.00i)9-s + (0.605 − 0.796i)11-s + (0.444 − 0.144i)13-s + (1.32 − 0.960i)15-s + (0.195 + 0.0636i)17-s + (0.804 − 1.10i)19-s − 0.395i·21-s + 0.937·23-s + (−0.242 − 0.175i)25-s + (0.0246 − 0.0759i)27-s + (0.424 + 0.583i)29-s + (0.529 + 1.62i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.09181 - 0.172794i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09181 - 0.172794i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (-73.2 + 96.3i)T \) |
| good | 3 | \( 1 + (10.4 + 7.58i)T + (25.0 + 77.0i)T^{2} \) |
| 5 | \( 1 + (8.80 - 27.1i)T + (-505. - 367. i)T^{2} \) |
| 7 | \( 1 + (-7.94 - 10.9i)T + (-741. + 2.28e3i)T^{2} \) |
| 13 | \( 1 + (-75.1 + 24.4i)T + (2.31e4 - 1.67e4i)T^{2} \) |
| 17 | \( 1 + (-56.6 - 18.3i)T + (6.75e4 + 4.90e4i)T^{2} \) |
| 19 | \( 1 + (-290. + 399. i)T + (-4.02e4 - 1.23e5i)T^{2} \) |
| 23 | \( 1 - 496.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-356. - 490. i)T + (-2.18e5 + 6.72e5i)T^{2} \) |
| 31 | \( 1 + (-508. - 1.56e3i)T + (-7.47e5 + 5.42e5i)T^{2} \) |
| 37 | \( 1 + (-1.15e3 + 841. i)T + (5.79e5 - 1.78e6i)T^{2} \) |
| 41 | \( 1 + (1.01e3 - 1.39e3i)T + (-8.73e5 - 2.68e6i)T^{2} \) |
| 43 | \( 1 + 3.02e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (77.1 + 56.0i)T + (1.50e6 + 4.64e6i)T^{2} \) |
| 53 | \( 1 + (-1.01e3 - 3.12e3i)T + (-6.38e6 + 4.63e6i)T^{2} \) |
| 59 | \( 1 + (-4.08e3 + 2.96e3i)T + (3.74e6 - 1.15e7i)T^{2} \) |
| 61 | \( 1 + (1.14e3 + 372. i)T + (1.12e7 + 8.13e6i)T^{2} \) |
| 67 | \( 1 + 6.90e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (176. - 543. i)T + (-2.05e7 - 1.49e7i)T^{2} \) |
| 73 | \( 1 + (-5.18e3 - 7.13e3i)T + (-8.77e6 + 2.70e7i)T^{2} \) |
| 79 | \( 1 + (-1.78e3 + 581. i)T + (3.15e7 - 2.28e7i)T^{2} \) |
| 83 | \( 1 + (6.04e3 + 1.96e3i)T + (3.83e7 + 2.78e7i)T^{2} \) |
| 89 | \( 1 - 1.21e4T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-307. - 946. i)T + (-7.16e7 + 5.20e7i)T^{2} \) |
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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
−13.28196806070122984271214605270, −12.02982158229680233046649402999, −11.32164120131486422060839070586, −10.61950263903421345379714434247, −8.785938347948218667237612877525, −7.19785627630653673307054511885, −6.54930066085038660406880653883, −5.30640970864101972585760531525, −3.17221824975389686244448840855, −0.943669735052710062307651294428,
0.966633702534743138011062802622, 4.10176085731166184912748271315, 4.92295610551931046697098472940, 6.15967438273591159078768037804, 7.85995899546052161044114494639, 9.301257206672752249512508029259, 10.20938145241434988414015575839, 11.52439657792887626377393338477, 12.06049793774447081242154194469, 13.27931466576419805890395111286
