| L(s) = 1 | + 1.68·3-s + 0.420·5-s − 0.389·7-s − 0.146·9-s − 3.59·11-s + 5.09·13-s + 0.710·15-s − 4.08·17-s − 1.64·19-s − 0.658·21-s − 6.76·23-s − 4.82·25-s − 5.31·27-s + 9.75·29-s − 4.10·31-s − 6.07·33-s − 0.163·35-s − 0.0847·37-s + 8.60·39-s − 9.32·41-s + 6.89·43-s − 0.0614·45-s − 7.43·47-s − 6.84·49-s − 6.89·51-s − 0.986·53-s − 1.51·55-s + ⋯ |
| L(s) = 1 | + 0.975·3-s + 0.188·5-s − 0.147·7-s − 0.0487·9-s − 1.08·11-s + 1.41·13-s + 0.183·15-s − 0.990·17-s − 0.377·19-s − 0.143·21-s − 1.41·23-s − 0.964·25-s − 1.02·27-s + 1.81·29-s − 0.737·31-s − 1.05·33-s − 0.0277·35-s − 0.0139·37-s + 1.37·39-s − 1.45·41-s + 1.05·43-s − 0.00915·45-s − 1.08·47-s − 0.978·49-s − 0.966·51-s − 0.135·53-s − 0.203·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 1.68T + 3T^{2} \) |
| 5 | \( 1 - 0.420T + 5T^{2} \) |
| 7 | \( 1 + 0.389T + 7T^{2} \) |
| 11 | \( 1 + 3.59T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 19 | \( 1 + 1.64T + 19T^{2} \) |
| 23 | \( 1 + 6.76T + 23T^{2} \) |
| 29 | \( 1 - 9.75T + 29T^{2} \) |
| 31 | \( 1 + 4.10T + 31T^{2} \) |
| 37 | \( 1 + 0.0847T + 37T^{2} \) |
| 41 | \( 1 + 9.32T + 41T^{2} \) |
| 43 | \( 1 - 6.89T + 43T^{2} \) |
| 47 | \( 1 + 7.43T + 47T^{2} \) |
| 53 | \( 1 + 0.986T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 8.89T + 67T^{2} \) |
| 71 | \( 1 - 2.68T + 71T^{2} \) |
| 73 | \( 1 - 9.46T + 73T^{2} \) |
| 79 | \( 1 + 4.71T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−8.267009736752406263671844165740, −7.63148855309525221823913095070, −6.46962092925144783486941788308, −6.03675648730561205001674202409, −5.04096611852111823776027036014, −4.06197691073275482021932654504, −3.35149896101831109245089683651, −2.48306317950510398312398647289, −1.73950682364969971130318672364, 0,
1.73950682364969971130318672364, 2.48306317950510398312398647289, 3.35149896101831109245089683651, 4.06197691073275482021932654504, 5.04096611852111823776027036014, 6.03675648730561205001674202409, 6.46962092925144783486941788308, 7.63148855309525221823913095070, 8.267009736752406263671844165740
