| L(s) = 1 | + (−2.72 + 1.26i)3-s + (0.689 + 4.95i)5-s − 0.735i·7-s + (5.82 − 6.86i)9-s + 10.9i·11-s − 21.1i·13-s + (−8.11 − 12.6i)15-s − 7.03·17-s − 23.1·19-s + (0.927 + 2.00i)21-s + 24.7·23-s + (−24.0 + 6.82i)25-s + (−7.21 + 26.0i)27-s − 32.3i·29-s − 34.9·31-s + ⋯ |
| L(s) = 1 | + (−0.907 + 0.420i)3-s + (0.137 + 0.990i)5-s − 0.105i·7-s + (0.647 − 0.762i)9-s + 0.995i·11-s − 1.63i·13-s + (−0.541 − 0.840i)15-s − 0.413·17-s − 1.21·19-s + (0.0441 + 0.0953i)21-s + 1.07·23-s + (−0.962 + 0.272i)25-s + (−0.267 + 0.963i)27-s − 1.11i·29-s − 1.12·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7712155511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7712155511\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.72 - 1.26i)T \) |
| 5 | \( 1 + (-0.689 - 4.95i)T \) |
| good | 7 | \( 1 + 0.735iT - 49T^{2} \) |
| 11 | \( 1 - 10.9iT - 121T^{2} \) |
| 13 | \( 1 + 21.1iT - 169T^{2} \) |
| 17 | \( 1 + 7.03T + 289T^{2} \) |
| 19 | \( 1 + 23.1T + 361T^{2} \) |
| 23 | \( 1 - 24.7T + 529T^{2} \) |
| 29 | \( 1 + 32.3iT - 841T^{2} \) |
| 31 | \( 1 + 34.9T + 961T^{2} \) |
| 37 | \( 1 + 37.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 39.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 22.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 39.1T + 2.20e3T^{2} \) |
| 53 | \( 1 - 60.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 7.79iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 11.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 33.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 96.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 134. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 121.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 90.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 53.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 115. iT - 9.40e3T^{2} \) |
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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
−9.976264610753892117251115948671, −9.106801583937796029690352139293, −7.78083552457852978844856673970, −7.04801390657395575408939136896, −6.23479928868930813548241773331, −5.42400922093691857112055019271, −4.42169135860755735286952038049, −3.40445669812482647241584344743, −2.14377796849477621941997307018, −0.31634017418413208575156666394,
1.07502980981406236658271039313, 2.12404842130682739947658665628, 3.95179620923138369543091524372, 4.82604694614038341780141203845, 5.62361343630633388970562501764, 6.52861582393508864112828291095, 7.20266889151087458479169558000, 8.622260268413083098996936235176, 8.837009602207884279521936491212, 10.00546291568044938504974197193
