| L(s) = 1 | + 4.18i·5-s + 2.44·7-s + 4.24i·11-s + 0.717i·13-s − 3·17-s − 6.24i·19-s + 1.01·23-s − 12.4·25-s + 4.18i·29-s − 8.36·31-s + 10.2i·35-s − 7.64i·37-s − 8.48·41-s + 12.2i·43-s + 8.36·47-s + ⋯ |
| L(s) = 1 | + 1.87i·5-s + 0.925·7-s + 1.27i·11-s + 0.198i·13-s − 0.727·17-s − 1.43i·19-s + 0.211·23-s − 2.49·25-s + 0.776i·29-s − 1.50·31-s + 1.73i·35-s − 1.25i·37-s − 1.32·41-s + 1.86i·43-s + 1.21·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.064296763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.064296763\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4.18iT - 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 - 4.24iT - 11T^{2} \) |
| 13 | \( 1 - 0.717iT - 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 6.24iT - 19T^{2} \) |
| 23 | \( 1 - 1.01T + 23T^{2} \) |
| 29 | \( 1 - 4.18iT - 29T^{2} \) |
| 31 | \( 1 + 8.36T + 31T^{2} \) |
| 37 | \( 1 + 7.64iT - 37T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 - 12.2iT - 43T^{2} \) |
| 47 | \( 1 - 8.36T + 47T^{2} \) |
| 53 | \( 1 + 2.02iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 5.61iT - 61T^{2} \) |
| 67 | \( 1 + 2.24iT - 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 + 5.91T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 - 0.485T + 97T^{2} \) |
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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
−8.589164071952561210769163261469, −7.51547959800060799753375650621, −7.16367581638908367362241382124, −6.75270615790238362965559995091, −5.78677099813344017038124773654, −4.85034558421878409413568659766, −4.19470896289395784590780219821, −3.19502396261101188935396118553, −2.38005729095416767001629861348, −1.75484249616904476460695898422,
0.26837521967995722352369452448, 1.31826761141512904990608328331, 1.99067990121863606985139768298, 3.49070447177960266955384565296, 4.18656063641864105689161731079, 4.96924036454590091108474808922, 5.54761562327151687315236344705, 6.06930441451264998529834644250, 7.33795759935089482731661699225, 8.095603177478994471196738490015
