| L(s) = 1 | + (−4.89 − 2.82i)2-s + (41.4 + 23.9i)3-s + (15.9 + 27.7i)4-s + (94.3 − 163. i)5-s + (−135. − 234. i)6-s + 72.4·7-s − 181. i·8-s + (778. + 1.34e3i)9-s + (−923. + 533. i)10-s − 1.29e3·11-s + 1.52e3i·12-s + (3.27e3 − 1.88e3i)13-s + (−355. − 204. i)14-s + (7.80e3 − 4.50e3i)15-s + (−512. + 886. i)16-s + (−2.49e3 + 4.31e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (1.53 + 0.885i)3-s + (0.249 + 0.433i)4-s + (0.754 − 1.30i)5-s + (−0.625 − 1.08i)6-s + 0.211·7-s − 0.353i·8-s + (1.06 + 1.84i)9-s + (−0.923 + 0.533i)10-s − 0.972·11-s + 0.885i·12-s + (1.48 − 0.859i)13-s + (−0.129 − 0.0746i)14-s + (2.31 − 1.33i)15-s + (−0.125 + 0.216i)16-s + (−0.507 + 0.879i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.32811 - 0.182242i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.32811 - 0.182242i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.89 + 2.82i)T \) |
| 19 | \( 1 + (-4.81e3 + 4.88e3i)T \) |
| good | 3 | \( 1 + (-41.4 - 23.9i)T + (364.5 + 631. i)T^{2} \) |
| 5 | \( 1 + (-94.3 + 163. i)T + (-7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 - 72.4T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.29e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + (-3.27e3 + 1.88e3i)T + (2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 + (2.49e3 - 4.31e3i)T + (-1.20e7 - 2.09e7i)T^{2} \) |
| 23 | \( 1 + (-7.09e3 - 1.22e4i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (1.78e4 - 1.03e4i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + 4.27e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 5.01e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (9.16e4 + 5.28e4i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (6.03e3 - 1.04e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (5.47e4 + 9.47e4i)T + (-5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (1.18e5 - 6.84e4i)T + (1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-1.82e4 - 1.05e4i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.00e5 + 1.73e5i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-2.44e5 + 1.41e5i)T + (4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + (1.79e5 + 1.03e5i)T + (6.40e10 + 1.10e11i)T^{2} \) |
| 73 | \( 1 + (1.57e5 - 2.72e5i)T + (-7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (9.72e4 + 5.61e4i)T + (1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 - 4.57e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + (9.28e5 - 5.35e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + (-8.49e5 - 4.90e5i)T + (4.16e11 + 7.21e11i)T^{2} \) |
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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
−15.31552335225974216024117971964, −13.44579072200353116979558821127, −13.15201024155879417620026708357, −10.82900378859406168226681703408, −9.690545419508884478866349560606, −8.752597982169531938471135736484, −8.081668208729971505968061778516, −5.12828149895377828954862998941, −3.35404829297663812523943247791, −1.61539360929056145258913848708,
1.78381567047040756663599643534, 3.01457383786774659377733268149, 6.37334730589928923897938832131, 7.36845919881133792490250635646, 8.555388479231920638130278166417, 9.735132757825282377077409088186, 11.12277371906306071919833583693, 13.24854373131133181818512728077, 14.02097766689226972738825801166, 14.76470763797130433419525017629
