| L(s) = 1 | + 1.25·3-s + 3.93·5-s − 3.56·7-s − 1.42·9-s − 3.07·11-s + 2.74·13-s + 4.94·15-s − 17-s + 2.16·19-s − 4.47·21-s − 5.55·23-s + 10.4·25-s − 5.55·27-s − 2.25·29-s − 0.00422·31-s − 3.86·33-s − 14.0·35-s + 5.75·37-s + 3.45·39-s − 2.70·41-s − 11.1·43-s − 5.59·45-s + 3.90·47-s + 5.70·49-s − 1.25·51-s − 6.71·53-s − 12.1·55-s + ⋯ |
| L(s) = 1 | + 0.725·3-s + 1.75·5-s − 1.34·7-s − 0.473·9-s − 0.928·11-s + 0.762·13-s + 1.27·15-s − 0.242·17-s + 0.496·19-s − 0.977·21-s − 1.15·23-s + 2.09·25-s − 1.06·27-s − 0.419·29-s − 0.000758·31-s − 0.673·33-s − 2.37·35-s + 0.946·37-s + 0.553·39-s − 0.422·41-s − 1.69·43-s − 0.833·45-s + 0.569·47-s + 0.814·49-s − 0.175·51-s − 0.922·53-s − 1.63·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 1.25T + 3T^{2} \) |
| 5 | \( 1 - 3.93T + 5T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 + 3.07T + 11T^{2} \) |
| 13 | \( 1 - 2.74T + 13T^{2} \) |
| 19 | \( 1 - 2.16T + 19T^{2} \) |
| 23 | \( 1 + 5.55T + 23T^{2} \) |
| 29 | \( 1 + 2.25T + 29T^{2} \) |
| 31 | \( 1 + 0.00422T + 31T^{2} \) |
| 37 | \( 1 - 5.75T + 37T^{2} \) |
| 41 | \( 1 + 2.70T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 3.90T + 47T^{2} \) |
| 53 | \( 1 + 6.71T + 53T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 5.44T + 67T^{2} \) |
| 71 | \( 1 + 3.00T + 71T^{2} \) |
| 73 | \( 1 + 0.207T + 73T^{2} \) |
| 79 | \( 1 + 3.57T + 79T^{2} \) |
| 83 | \( 1 + 4.16T + 83T^{2} \) |
| 89 | \( 1 - 5.88T + 89T^{2} \) |
| 97 | \( 1 + 9.10T + 97T^{2} \) |
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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
−7.52788978205686659760686879770, −6.59398927872605937866585336157, −5.98358238128916350385301715642, −5.73831318444347144639312799020, −4.77582833614876706353809958604, −3.55353037819134240641169787807, −2.98562660362467806355966200537, −2.36560868754025837701214428061, −1.55335398052620210537217118422, 0,
1.55335398052620210537217118422, 2.36560868754025837701214428061, 2.98562660362467806355966200537, 3.55353037819134240641169787807, 4.77582833614876706353809958604, 5.73831318444347144639312799020, 5.98358238128916350385301715642, 6.59398927872605937866585336157, 7.52788978205686659760686879770
