| L(s) = 1 | − 5-s − 1.44·7-s − 3.44·11-s + 3.89·13-s + 4.89·17-s − 4·19-s + 0.550·23-s − 4·25-s + 9.89·29-s + 7.44·31-s + 1.44·35-s − 8.89·37-s + 2.10·41-s − 12.3·43-s − 8.34·47-s − 4.89·49-s − 0.898·53-s + 3.44·55-s − 0.348·59-s − 1.89·61-s − 3.89·65-s + 2.34·67-s + 11.7·71-s + 4.89·73-s + 5·77-s + 8.55·79-s − 5.44·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.547·7-s − 1.04·11-s + 1.08·13-s + 1.18·17-s − 0.917·19-s + 0.114·23-s − 0.800·25-s + 1.83·29-s + 1.33·31-s + 0.245·35-s − 1.46·37-s + 0.328·41-s − 1.88·43-s − 1.21·47-s − 0.699·49-s − 0.123·53-s + 0.465·55-s − 0.0453·59-s − 0.243·61-s − 0.483·65-s + 0.286·67-s + 1.40·71-s + 0.573·73-s + 0.569·77-s + 0.962·79-s − 0.598·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.376466090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376466090\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + 5T^{2} \) |
| 7 | \( 1 + 1.44T + 7T^{2} \) |
| 11 | \( 1 + 3.44T + 11T^{2} \) |
| 13 | \( 1 - 3.89T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 0.550T + 23T^{2} \) |
| 29 | \( 1 - 9.89T + 29T^{2} \) |
| 31 | \( 1 - 7.44T + 31T^{2} \) |
| 37 | \( 1 + 8.89T + 37T^{2} \) |
| 41 | \( 1 - 2.10T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 + 8.34T + 47T^{2} \) |
| 53 | \( 1 + 0.898T + 53T^{2} \) |
| 59 | \( 1 + 0.348T + 59T^{2} \) |
| 61 | \( 1 + 1.89T + 61T^{2} \) |
| 67 | \( 1 - 2.34T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 - 8.55T + 79T^{2} \) |
| 83 | \( 1 + 5.44T + 83T^{2} \) |
| 89 | \( 1 - 3.10T + 89T^{2} \) |
| 97 | \( 1 - 5.89T + 97T^{2} \) |
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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
−8.350648738054536490058110367682, −7.64287355196005200171453474951, −6.59137272344676924696540817606, −6.24891746570587115534306671466, −5.23585467408284324446555680891, −4.58666035663499277059827581615, −3.48942236621643423563232674508, −3.10539745667016071057988596127, −1.88948321060473547634499022827, −0.62810277698024323352888886112,
0.62810277698024323352888886112, 1.88948321060473547634499022827, 3.10539745667016071057988596127, 3.48942236621643423563232674508, 4.58666035663499277059827581615, 5.23585467408284324446555680891, 6.24891746570587115534306671466, 6.59137272344676924696540817606, 7.64287355196005200171453474951, 8.350648738054536490058110367682
