| L(s) = 1 | − 0.950·3-s − 2.28·5-s − 2.71·7-s − 2.09·9-s − 1.36·11-s − 2.93·13-s + 2.17·15-s − 2.61·17-s + 7.79·19-s + 2.57·21-s + 2.61·23-s + 0.230·25-s + 4.84·27-s − 0.314·29-s + 7.95·31-s + 1.30·33-s + 6.19·35-s − 4.17·37-s + 2.78·39-s + 6.16·41-s − 0.457·43-s + 4.79·45-s − 7.67·47-s + 0.345·49-s + 2.48·51-s + 7.26·53-s + 3.13·55-s + ⋯ |
| L(s) = 1 | − 0.548·3-s − 1.02·5-s − 1.02·7-s − 0.699·9-s − 0.412·11-s − 0.812·13-s + 0.561·15-s − 0.633·17-s + 1.78·19-s + 0.561·21-s + 0.544·23-s + 0.0460·25-s + 0.932·27-s − 0.0583·29-s + 1.42·31-s + 0.226·33-s + 1.04·35-s − 0.686·37-s + 0.445·39-s + 0.962·41-s − 0.0698·43-s + 0.715·45-s − 1.11·47-s + 0.0493·49-s + 0.347·51-s + 0.997·53-s + 0.422·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 0.950T + 3T^{2} \) |
| 5 | \( 1 + 2.28T + 5T^{2} \) |
| 7 | \( 1 + 2.71T + 7T^{2} \) |
| 11 | \( 1 + 1.36T + 11T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 + 2.61T + 17T^{2} \) |
| 19 | \( 1 - 7.79T + 19T^{2} \) |
| 23 | \( 1 - 2.61T + 23T^{2} \) |
| 29 | \( 1 + 0.314T + 29T^{2} \) |
| 31 | \( 1 - 7.95T + 31T^{2} \) |
| 37 | \( 1 + 4.17T + 37T^{2} \) |
| 41 | \( 1 - 6.16T + 41T^{2} \) |
| 43 | \( 1 + 0.457T + 43T^{2} \) |
| 47 | \( 1 + 7.67T + 47T^{2} \) |
| 53 | \( 1 - 7.26T + 53T^{2} \) |
| 59 | \( 1 + 0.217T + 59T^{2} \) |
| 61 | \( 1 - 7.26T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 16.5T + 71T^{2} \) |
| 73 | \( 1 - 2.86T + 73T^{2} \) |
| 79 | \( 1 - 16.0T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 0.0952T + 89T^{2} \) |
| 97 | \( 1 + 9.27T + 97T^{2} \) |
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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
−7.44335898013687975263001788768, −6.80635525327280845894496646744, −6.17139431624754729187976847829, −5.27676297461432356945986576816, −4.84521580484567033565388552192, −3.81113690977989338372067634145, −3.11322080480123855014306698035, −2.51012258927727108144791064871, −0.831289083494322303812140328840, 0,
0.831289083494322303812140328840, 2.51012258927727108144791064871, 3.11322080480123855014306698035, 3.81113690977989338372067634145, 4.84521580484567033565388552192, 5.27676297461432356945986576816, 6.17139431624754729187976847829, 6.80635525327280845894496646744, 7.44335898013687975263001788768
