| L(s) = 1 | + 3.37·5-s + 1.37·7-s + 11-s − 5.37·13-s + 0.372·17-s + 6.37·19-s − 5.37·23-s + 6.37·25-s − 1.37·29-s − 0.627·31-s + 4.62·35-s + 2.74·37-s − 0.255·41-s + 9.74·43-s + 1.37·47-s − 5.11·49-s + 10.7·53-s + 3.37·55-s + 7·59-s + 3.37·61-s − 18.1·65-s + 7.74·67-s + 4·71-s + 5.11·73-s + 1.37·77-s + 0.627·79-s + 15.3·83-s + ⋯ |
| L(s) = 1 | + 1.50·5-s + 0.518·7-s + 0.301·11-s − 1.49·13-s + 0.0902·17-s + 1.46·19-s − 1.12·23-s + 1.27·25-s − 0.254·29-s − 0.112·31-s + 0.782·35-s + 0.451·37-s − 0.0398·41-s + 1.48·43-s + 0.200·47-s − 0.730·49-s + 1.47·53-s + 0.454·55-s + 0.911·59-s + 0.431·61-s − 2.24·65-s + 0.946·67-s + 0.474·71-s + 0.598·73-s + 0.156·77-s + 0.0706·79-s + 1.68·83-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
| \(L(1)\) |
\(\approx\) |
\(2.876938579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.876938579\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.37T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 5.37T + 13T^{2} \) |
| 17 | \( 1 - 0.372T + 17T^{2} \) |
| 19 | \( 1 - 6.37T + 19T^{2} \) |
| 23 | \( 1 + 5.37T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 + 0.627T + 31T^{2} \) |
| 37 | \( 1 - 2.74T + 37T^{2} \) |
| 41 | \( 1 + 0.255T + 41T^{2} \) |
| 43 | \( 1 - 9.74T + 43T^{2} \) |
| 47 | \( 1 - 1.37T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 7T + 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 - 7.74T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 5.11T + 73T^{2} \) |
| 79 | \( 1 - 0.627T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 9.74T + 97T^{2} \) |
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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
−8.138629231776818740532688947296, −7.46624840873072606194571742725, −6.79010531190910557146903464271, −5.87388744257627378641659435332, −5.39104718504408689768664076557, −4.75408885331378748183654761453, −3.70762889340938545267862313589, −2.51315983834014644421953462582, −2.05473703962203636975436807099, −0.948106115100228775667493297987,
0.948106115100228775667493297987, 2.05473703962203636975436807099, 2.51315983834014644421953462582, 3.70762889340938545267862313589, 4.75408885331378748183654761453, 5.39104718504408689768664076557, 5.87388744257627378641659435332, 6.79010531190910557146903464271, 7.46624840873072606194571742725, 8.138629231776818740532688947296
