| L(s) = 1 | + 2-s − 2.07·3-s + 4-s + 2.52·5-s − 2.07·6-s + 3.77·7-s + 8-s + 1.31·9-s + 2.52·10-s − 0.951·11-s − 2.07·12-s + 3.77·14-s − 5.25·15-s + 16-s + 6.99·17-s + 1.31·18-s + 19-s + 2.52·20-s − 7.83·21-s − 0.951·22-s − 1.25·23-s − 2.07·24-s + 1.39·25-s + 3.50·27-s + 3.77·28-s + 0.0699·29-s − 5.25·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.19·3-s + 0.5·4-s + 1.13·5-s − 0.847·6-s + 1.42·7-s + 0.353·8-s + 0.437·9-s + 0.799·10-s − 0.286·11-s − 0.599·12-s + 1.00·14-s − 1.35·15-s + 0.250·16-s + 1.69·17-s + 0.309·18-s + 0.229·19-s + 0.565·20-s − 1.71·21-s − 0.202·22-s − 0.262·23-s − 0.423·24-s + 0.278·25-s + 0.674·27-s + 0.713·28-s + 0.0129·29-s − 0.958·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.420165665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.420165665\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.07T + 3T^{2} \) |
| 5 | \( 1 - 2.52T + 5T^{2} \) |
| 7 | \( 1 - 3.77T + 7T^{2} \) |
| 11 | \( 1 + 0.951T + 11T^{2} \) |
| 17 | \( 1 - 6.99T + 17T^{2} \) |
| 23 | \( 1 + 1.25T + 23T^{2} \) |
| 29 | \( 1 - 0.0699T + 29T^{2} \) |
| 31 | \( 1 - 9.19T + 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 + 4.57T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 3.75T + 47T^{2} \) |
| 53 | \( 1 - 6.49T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 1.73T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 9.05T + 71T^{2} \) |
| 73 | \( 1 - 3.72T + 73T^{2} \) |
| 79 | \( 1 + 5.62T + 79T^{2} \) |
| 83 | \( 1 + 2.15T + 83T^{2} \) |
| 89 | \( 1 + 1.02T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−8.064816932521443159702605770899, −7.01855118272368280478967632479, −6.38239200411189230090521275134, −5.63740741463299937325970070308, −5.19628980647219759942314646610, −4.91417123374689591882770359588, −3.77479744087809677253427571901, −2.68871640134510177943594105768, −1.74005533653167717166438981282, −0.995662284877952770181673669208,
0.995662284877952770181673669208, 1.74005533653167717166438981282, 2.68871640134510177943594105768, 3.77479744087809677253427571901, 4.91417123374689591882770359588, 5.19628980647219759942314646610, 5.63740741463299937325970070308, 6.38239200411189230090521275134, 7.01855118272368280478967632479, 8.064816932521443159702605770899
