| L(s) = 1 | + (−14.8 − 5.93i)2-s + (40.5 + 23.3i)3-s + (185. + 176. i)4-s + (−333. − 577. i)5-s + (−463. − 587. i)6-s + (−2.38e3 − 255. i)7-s + (−1.71e3 − 3.72e3i)8-s + (1.09e3 + 1.89e3i)9-s + (1.52e3 + 1.05e4i)10-s + (−2.11e4 − 1.21e4i)11-s + (3.39e3 + 1.14e4i)12-s + 3.60e4·13-s + (3.39e4 + 1.79e4i)14-s − 3.11e4i·15-s + (3.35e3 + 6.54e4i)16-s + (−3.30e4 + 5.72e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.928 − 0.370i)2-s + (0.5 + 0.288i)3-s + (0.725 + 0.688i)4-s + (−0.533 − 0.923i)5-s + (−0.357 − 0.453i)6-s + (−0.994 − 0.106i)7-s + (−0.417 − 0.908i)8-s + (0.166 + 0.288i)9-s + (0.152 + 1.05i)10-s + (−1.44 − 0.832i)11-s + (0.163 + 0.553i)12-s + 1.26·13-s + (0.883 + 0.467i)14-s − 0.615i·15-s + (0.0512 + 0.998i)16-s + (−0.396 + 0.685i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.456286 + 0.300426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.456286 + 0.300426i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (14.8 + 5.93i)T \) |
| 3 | \( 1 + (-40.5 - 23.3i)T \) |
| 7 | \( 1 + (2.38e3 + 255. i)T \) |
| good | 5 | \( 1 + (333. + 577. i)T + (-1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (2.11e4 + 1.21e4i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 3.60e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (3.30e4 - 5.72e4i)T + (-3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.40e5 + 8.12e4i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (2.53e5 - 1.46e5i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 2.52e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + (9.34e5 + 5.39e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-1.22e6 - 2.12e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 1.23e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 6.46e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-5.23e6 + 3.02e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (3.60e6 - 6.24e6i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-6.99e6 - 4.04e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-5.86e6 - 1.01e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (2.58e7 + 1.49e7i)T + (2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 3.85e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (8.04e6 - 1.39e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (5.62e5 - 3.24e5i)T + (7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 - 1.63e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-4.96e6 - 8.59e6i)T + (-1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.12e8T + 7.83e15T^{2} \) |
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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
−12.95855490425585004721121181520, −11.51475455338817196438602785386, −10.50361587963569722023120997323, −9.370731805937964530579294053550, −8.477159137280229461478264137338, −7.66660399264861644772649404379, −5.91960582147363206282202275459, −3.92485319747670227372183217893, −2.84187548687163955045646961020, −0.969567194760037749632830404840,
0.25358629060272645701129067941, 2.22069418047597350120720985480, 3.39093062354362242212589247968, 5.78182509422529538779325766769, 7.07602437415285232193430194577, 7.69217357747498028309597814718, 9.025687103811256214522026012748, 10.13168304836624272743926529561, 10.98485383043511200162347590384, 12.37082686159840054545474472912
