| L(s) = 1 | + (−43.8 + 15.9i)3-s + (−73.2 + 415. i)5-s + (823. + 1.42e3i)7-s + (−6.52 + 5.47i)9-s + (489. − 848. i)11-s + (1.18e4 + 4.30e3i)13-s + (−3.41e3 − 1.93e4i)15-s + (2.10e3 + 1.76e3i)17-s + (−1.49e4 + 2.58e4i)19-s + (−5.88e4 − 4.93e4i)21-s + (1.90e4 + 1.08e5i)23-s + (−9.38e4 − 3.41e4i)25-s + (5.12e4 − 8.87e4i)27-s + (−9.71e4 + 8.15e4i)29-s + (−1.08e5 − 1.87e5i)31-s + ⋯ |
| L(s) = 1 | + (−0.937 + 0.341i)3-s + (−0.262 + 1.48i)5-s + (0.907 + 1.57i)7-s + (−0.00298 + 0.00250i)9-s + (0.110 − 0.192i)11-s + (1.49 + 0.544i)13-s + (−0.261 − 1.48i)15-s + (0.104 + 0.0873i)17-s + (−0.499 + 0.866i)19-s + (−1.38 − 1.16i)21-s + (0.326 + 1.85i)23-s + (−1.20 − 0.437i)25-s + (0.500 − 0.867i)27-s + (−0.739 + 0.620i)29-s + (−0.651 − 1.12i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0152i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.00994857 - 1.30404i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00994857 - 1.30404i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.49e4 - 2.58e4i)T \) |
| good | 3 | \( 1 + (43.8 - 15.9i)T + (1.67e3 - 1.40e3i)T^{2} \) |
| 5 | \( 1 + (73.2 - 415. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-823. - 1.42e3i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-489. + 848. i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-1.18e4 - 4.30e3i)T + (4.80e7 + 4.03e7i)T^{2} \) |
| 17 | \( 1 + (-2.10e3 - 1.76e3i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 23 | \( 1 + (-1.90e4 - 1.08e5i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (9.71e4 - 8.15e4i)T + (2.99e9 - 1.69e10i)T^{2} \) |
| 31 | \( 1 + (1.08e5 + 1.87e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 2.62e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.52e5 + 5.54e4i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (-1.25e5 + 7.09e5i)T + (-2.55e11 - 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-3.70e5 + 3.11e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + (-7.38e4 - 4.18e5i)T + (-1.10e12 + 4.01e11i)T^{2} \) |
| 59 | \( 1 + (7.36e5 + 6.18e5i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (-1.03e5 - 5.85e5i)T + (-2.95e12 + 1.07e12i)T^{2} \) |
| 67 | \( 1 + (1.88e5 - 1.58e5i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (-9.19e5 + 5.21e6i)T + (-8.54e12 - 3.11e12i)T^{2} \) |
| 73 | \( 1 + (-2.98e6 + 1.08e6i)T + (8.46e12 - 7.10e12i)T^{2} \) |
| 79 | \( 1 + (-6.19e6 + 2.25e6i)T + (1.47e13 - 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-2.15e6 - 3.73e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-5.76e6 - 2.09e6i)T + (3.38e13 + 2.84e13i)T^{2} \) |
| 97 | \( 1 + (-4.81e6 - 4.03e6i)T + (1.40e13 + 7.95e13i)T^{2} \) |
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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
−13.80261877918677803552692903693, −12.02369948109598843035109257258, −11.21478461535537914999521521512, −10.86272604320437392159307085499, −9.138292104100505572558873305322, −7.82106064643933236716869977868, −6.18086639884493770894654818496, −5.53388434418979986127217712000, −3.65333658845615099316404681731, −1.99506706273208855389528776224,
0.62319656372803995040811550576, 1.11286516501874231342534953105, 4.13240439936904119171457859235, 5.04118313317081220343963775401, 6.48259275377107548761808806231, 7.905318388667598964507670224916, 8.879373904786482166689460832644, 10.71868283633088403354441115877, 11.30251812209174092320708960822, 12.62673618550735039581341044448
