| L(s) = 1 | + 3.29i·5-s − 2.46·7-s − 10.4i·11-s + 2.93·13-s + 27.8i·17-s − 4.35·19-s + 24.3i·23-s + 14.1·25-s + 7.80i·29-s − 17.4·31-s − 8.11i·35-s − 48.3·37-s + 51.2i·41-s − 82.7·43-s − 22.2i·47-s + ⋯ |
| L(s) = 1 | + 0.659i·5-s − 0.351·7-s − 0.946i·11-s + 0.225·13-s + 1.63i·17-s − 0.229·19-s + 1.05i·23-s + 0.565·25-s + 0.269i·29-s − 0.562·31-s − 0.231i·35-s − 1.30·37-s + 1.24i·41-s − 1.92·43-s − 0.472i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.028223706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.028223706\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35T \) |
| good | 5 | \( 1 - 3.29iT - 25T^{2} \) |
| 7 | \( 1 + 2.46T + 49T^{2} \) |
| 11 | \( 1 + 10.4iT - 121T^{2} \) |
| 13 | \( 1 - 2.93T + 169T^{2} \) |
| 17 | \( 1 - 27.8iT - 289T^{2} \) |
| 23 | \( 1 - 24.3iT - 529T^{2} \) |
| 29 | \( 1 - 7.80iT - 841T^{2} \) |
| 31 | \( 1 + 17.4T + 961T^{2} \) |
| 37 | \( 1 + 48.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 51.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 82.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 22.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 16.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 49.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 16.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 1.05T + 4.48e3T^{2} \) |
| 71 | \( 1 - 79.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 53.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 137.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 4.56iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 120. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 13.4T + 9.40e3T^{2} \) |
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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
−10.65008732846234935815112501084, −9.848035034454392408125435367331, −8.740465899375545314772314326323, −8.094225015045343909263308290379, −6.89818291598668626334738998818, −6.23043213173808628875712878237, −5.28969619140412861107806429154, −3.78821880982289900694353232638, −3.11934220337638689486526870576, −1.56571743584623185652863554662,
0.36238574840059989202123130944, 1.95829754866115195545942750463, 3.28164930282459399405741795900, 4.61413115537447199399921992362, 5.20205915790930487869625359162, 6.56756512673050986937382939026, 7.23760182654263700092186249694, 8.373648316353537210232111955153, 9.149530921913736933459015614879, 9.872957036007929229964014939876
