| L(s) = 1 | + (0.205 − 1.42i)3-s + (−104. + 30.5i)5-s + (−109. − 126. i)7-s + (231. + 67.8i)9-s + (468. + 300. i)11-s + (640. − 738. i)13-s + (22.2 + 155. i)15-s + (762. + 1.66e3i)17-s + (−16.7 + 36.6i)19-s + (−202. + 130. i)21-s + (−2.21e3 − 1.23e3i)23-s + (7.28e3 − 4.68e3i)25-s + (290. − 635. i)27-s + (568. + 1.24e3i)29-s + (519. + 3.61e3i)31-s + ⋯ |
| L(s) = 1 | + (0.0131 − 0.0916i)3-s + (−1.86 + 0.547i)5-s + (−0.843 − 0.972i)7-s + (0.951 + 0.279i)9-s + (1.16 + 0.749i)11-s + (1.05 − 1.21i)13-s + (0.0255 + 0.177i)15-s + (0.639 + 1.40i)17-s + (−0.0106 + 0.0232i)19-s + (−0.100 + 0.0644i)21-s + (−0.872 − 0.487i)23-s + (2.33 − 1.49i)25-s + (0.0765 − 0.167i)27-s + (0.125 + 0.274i)29-s + (0.0971 + 0.675i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0211i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.30636 + 0.0138419i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30636 + 0.0138419i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (2.21e3 + 1.23e3i)T \) |
| good | 3 | \( 1 + (-0.205 + 1.42i)T + (-233. - 68.4i)T^{2} \) |
| 5 | \( 1 + (104. - 30.5i)T + (2.62e3 - 1.68e3i)T^{2} \) |
| 7 | \( 1 + (109. + 126. i)T + (-2.39e3 + 1.66e4i)T^{2} \) |
| 11 | \( 1 + (-468. - 300. i)T + (6.69e4 + 1.46e5i)T^{2} \) |
| 13 | \( 1 + (-640. + 738. i)T + (-5.28e4 - 3.67e5i)T^{2} \) |
| 17 | \( 1 + (-762. - 1.66e3i)T + (-9.29e5 + 1.07e6i)T^{2} \) |
| 19 | \( 1 + (16.7 - 36.6i)T + (-1.62e6 - 1.87e6i)T^{2} \) |
| 29 | \( 1 + (-568. - 1.24e3i)T + (-1.34e7 + 1.55e7i)T^{2} \) |
| 31 | \( 1 + (-519. - 3.61e3i)T + (-2.74e7 + 8.06e6i)T^{2} \) |
| 37 | \( 1 + (-2.05e3 - 602. i)T + (5.83e7 + 3.74e7i)T^{2} \) |
| 41 | \( 1 + (-5.79e3 + 1.70e3i)T + (9.74e7 - 6.26e7i)T^{2} \) |
| 43 | \( 1 + (-1.76e3 + 1.22e4i)T + (-1.41e8 - 4.14e7i)T^{2} \) |
| 47 | \( 1 - 1.77e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (3.18e3 + 3.67e3i)T + (-5.95e7 + 4.13e8i)T^{2} \) |
| 59 | \( 1 + (-7.04e3 + 8.13e3i)T + (-1.01e8 - 7.07e8i)T^{2} \) |
| 61 | \( 1 + (-2.15e3 - 1.49e4i)T + (-8.10e8 + 2.37e8i)T^{2} \) |
| 67 | \( 1 + (-3.57e4 + 2.29e4i)T + (5.60e8 - 1.22e9i)T^{2} \) |
| 71 | \( 1 + (-3.23e4 + 2.08e4i)T + (7.49e8 - 1.64e9i)T^{2} \) |
| 73 | \( 1 + (1.71e4 - 3.76e4i)T + (-1.35e9 - 1.56e9i)T^{2} \) |
| 79 | \( 1 + (4.13e4 - 4.76e4i)T + (-4.37e8 - 3.04e9i)T^{2} \) |
| 83 | \( 1 + (-1.02e4 - 3.01e3i)T + (3.31e9 + 2.12e9i)T^{2} \) |
| 89 | \( 1 + (1.64e4 - 1.14e5i)T + (-5.35e9 - 1.57e9i)T^{2} \) |
| 97 | \( 1 + (-8.11e4 + 2.38e4i)T + (7.22e9 - 4.64e9i)T^{2} \) |
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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
−12.80653151135812007547894008752, −12.22246963853905420931736529333, −10.78732429873552411217998569246, −10.19421235578724598753005232224, −8.323238660923036938962787014239, −7.39778411895354508161696732525, −6.53419789821950847852823500612, −4.04826550191508799573772793825, −3.65848405475102204938017143331, −0.878652919611415370364722331655,
0.862620232942329620137426373118, 3.48039367669819132561797405899, 4.31504247313583446473896950418, 6.23013390466170782622118423569, 7.45694471681466119760695663434, 8.792565950376566812556391386060, 9.451968258929096507075813062261, 11.53473207542838542240167763263, 11.77726299475951208071687173774, 12.83119663184158574702506166400
