| L(s) = 1 | + 3-s + 0.392·5-s − 0.0476·7-s + 9-s − 1.67·11-s − 4.70·13-s + 0.392·15-s + 0.835·17-s − 0.998·19-s − 0.0476·21-s − 1.31·23-s − 4.84·25-s + 27-s + 6.86·29-s + 9.48·31-s − 1.67·33-s − 0.0187·35-s + 2.44·37-s − 4.70·39-s + 0.448·41-s − 9.51·43-s + 0.392·45-s + 3.80·47-s − 6.99·49-s + 0.835·51-s − 9.39·53-s − 0.658·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.175·5-s − 0.0180·7-s + 0.333·9-s − 0.505·11-s − 1.30·13-s + 0.101·15-s + 0.202·17-s − 0.229·19-s − 0.0104·21-s − 0.273·23-s − 0.969·25-s + 0.192·27-s + 1.27·29-s + 1.70·31-s − 0.291·33-s − 0.00316·35-s + 0.402·37-s − 0.753·39-s + 0.0699·41-s − 1.45·43-s + 0.0584·45-s + 0.555·47-s − 0.999·49-s + 0.117·51-s − 1.29·53-s − 0.0887·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 0.392T + 5T^{2} \) |
| 7 | \( 1 + 0.0476T + 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 + 4.70T + 13T^{2} \) |
| 17 | \( 1 - 0.835T + 17T^{2} \) |
| 19 | \( 1 + 0.998T + 19T^{2} \) |
| 23 | \( 1 + 1.31T + 23T^{2} \) |
| 29 | \( 1 - 6.86T + 29T^{2} \) |
| 31 | \( 1 - 9.48T + 31T^{2} \) |
| 37 | \( 1 - 2.44T + 37T^{2} \) |
| 41 | \( 1 - 0.448T + 41T^{2} \) |
| 43 | \( 1 + 9.51T + 43T^{2} \) |
| 47 | \( 1 - 3.80T + 47T^{2} \) |
| 53 | \( 1 + 9.39T + 53T^{2} \) |
| 59 | \( 1 - 3.24T + 59T^{2} \) |
| 61 | \( 1 - 0.335T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 5.78T + 71T^{2} \) |
| 73 | \( 1 + 2.80T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 7.43T + 83T^{2} \) |
| 89 | \( 1 + 5.92T + 89T^{2} \) |
| 97 | \( 1 - 4.70T + 97T^{2} \) |
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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
−7.70716797686777658638675997528, −6.77628261993109248553034505314, −6.26196104092551159625364343497, −5.22605227465667452050781098153, −4.71087300827328452373387677765, −3.90870220512903852717793364030, −2.85197779656606871779096424306, −2.45918740501490198253170805797, −1.38646604510428597497458143228, 0,
1.38646604510428597497458143228, 2.45918740501490198253170805797, 2.85197779656606871779096424306, 3.90870220512903852717793364030, 4.71087300827328452373387677765, 5.22605227465667452050781098153, 6.26196104092551159625364343497, 6.77628261993109248553034505314, 7.70716797686777658638675997528
