L(s) = 1 | + (−4.89 + 2.82i)2-s + (18.9 − 10.9i)3-s + (15.9 − 27.7i)4-s + (−55.1 − 95.5i)5-s + (−61.8 + 107. i)6-s − 148.·7-s + 181. i·8-s + (−125. + 217. i)9-s + (540. + 311. i)10-s − 494.·11-s − 699. i·12-s + (−1.99e3 − 1.15e3i)13-s + (728. − 420. i)14-s + (−2.08e3 − 1.20e3i)15-s + (−512. − 886. i)16-s + (−2.82e3 − 4.88e3i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.701 − 0.404i)3-s + (0.249 − 0.433i)4-s + (−0.441 − 0.764i)5-s + (−0.286 + 0.496i)6-s − 0.433·7-s + 0.353i·8-s + (−0.171 + 0.297i)9-s + (0.540 + 0.311i)10-s − 0.371·11-s − 0.404i·12-s + (−0.906 − 0.523i)13-s + (0.265 − 0.153i)14-s + (−0.619 − 0.357i)15-s + (−0.125 − 0.216i)16-s + (−0.574 − 0.994i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.590i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.180592 - 0.553099i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.180592 - 0.553099i\) |
\(L(4)\) |
|
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 | \( 1 + (6.51e3 + 2.14e3i)T \) |
good | 3 | \( 1 + (-18.9 + 10.9i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (55.1 + 95.5i)T + (-7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + 148.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 494.T + 1.77e6T^{2} \) |
| 13 | \( 1 + (1.99e3 + 1.15e3i)T + (2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + (2.82e3 + 4.88e3i)T + (-1.20e7 + 2.09e7i)T^{2} \) |
| 23 | \( 1 + (-3.20e3 + 5.55e3i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-5.70e3 - 3.29e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 - 2.50e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 1.65e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-3.89e4 + 2.24e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (3.17e4 + 5.50e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-6.40e3 + 1.10e4i)T + (-5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.84e5 - 1.06e5i)T + (1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (-1.98e5 + 1.14e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-8.50e3 + 1.47e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.70e5 + 9.85e4i)T + (4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + (1.97e4 - 1.13e4i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (5.36e4 + 9.29e4i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-6.04e5 + 3.49e5i)T + (1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 8.75e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + (4.48e5 + 2.59e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (9.50e5 - 5.49e5i)T + (4.16e11 - 7.21e11i)T^{2} \) |
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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
−14.69068338882747646659294748798, −13.38347681392953428297655331865, −12.30734860593830485566864074160, −10.62841336869636865668261954983, −9.104873081563741722531046498843, −8.200955517315173677944393216832, −7.02642332504062769537286470671, −4.98420526111869745264185174414, −2.52292398991067705345825654411, −0.30121329566324430660212376901,
2.53929413407139905289595316375, 3.89553021322201212798628186362, 6.61539318418350635817956591549, 8.066114178731000604662451528728, 9.323583873492439719176880376678, 10.40389029362715105243093656641, 11.64212002649767664267189462054, 13.03880404177242973862852456649, 14.68656997237459005571071943612, 15.28472206657647491112058159359