| L(s) = 1 | + 3.37·3-s + 2.52·5-s − 3.31·7-s + 8.37·9-s + 2.37·11-s − 5.84·13-s + 8.51·15-s + 5·17-s − 19-s − 11.1·21-s − 0.792·23-s + 1.37·25-s + 18.1·27-s + 2.67·29-s + 3.46·31-s + 8·33-s − 8.37·35-s + 10.0·37-s − 19.6·39-s − 3.62·43-s + 21.1·45-s − 0.644·47-s + 4·49-s + 16.8·51-s + 6.13·53-s + 5.98·55-s − 3.37·57-s + ⋯ |
| L(s) = 1 | + 1.94·3-s + 1.12·5-s − 1.25·7-s + 2.79·9-s + 0.715·11-s − 1.61·13-s + 2.19·15-s + 1.21·17-s − 0.229·19-s − 2.44·21-s − 0.165·23-s + 0.274·25-s + 3.48·27-s + 0.496·29-s + 0.622·31-s + 1.39·33-s − 1.41·35-s + 1.65·37-s − 3.15·39-s − 0.553·43-s + 3.15·45-s − 0.0940·47-s + 0.571·49-s + 2.36·51-s + 0.842·53-s + 0.807·55-s − 0.446·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.796634136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.796634136\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 3.37T + 3T^{2} \) |
| 5 | \( 1 - 2.52T + 5T^{2} \) |
| 7 | \( 1 + 3.31T + 7T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 13 | \( 1 + 5.84T + 13T^{2} \) |
| 17 | \( 1 - 5T + 17T^{2} \) |
| 23 | \( 1 + 0.792T + 23T^{2} \) |
| 29 | \( 1 - 2.67T + 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 3.62T + 43T^{2} \) |
| 47 | \( 1 + 0.644T + 47T^{2} \) |
| 53 | \( 1 - 6.13T + 53T^{2} \) |
| 59 | \( 1 - 1.37T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 + 2.62T + 67T^{2} \) |
| 71 | \( 1 - 6.63T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 0.294T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 + 2.74T + 97T^{2} \) |
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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
−8.302107843454249583018874885906, −7.65929808528468218639895046861, −6.87940282784754454027124823131, −6.37573089069668888238124057017, −5.30783338121843965124774595369, −4.28395927871866492239782020689, −3.52012579712368695991915950300, −2.68830220642473865910340592535, −2.32234543097201405185811266083, −1.17578327672190567591310113795,
1.17578327672190567591310113795, 2.32234543097201405185811266083, 2.68830220642473865910340592535, 3.52012579712368695991915950300, 4.28395927871866492239782020689, 5.30783338121843965124774595369, 6.37573089069668888238124057017, 6.87940282784754454027124823131, 7.65929808528468218639895046861, 8.302107843454249583018874885906
