| L(s) = 1 | + 2.23·5-s + 7-s − 2.85·11-s − 3·13-s + 1.61·17-s − 19-s + 1.76·23-s + 0.854·29-s − 4.38·31-s + 2.23·35-s − 4.70·37-s + 1.14·41-s − 3.52·43-s + 7.47·47-s + 49-s − 10.3·53-s − 6.38·55-s + 3·59-s − 15.1·61-s − 6.70·65-s + 2.85·67-s + 15.1·71-s + 2.32·73-s − 2.85·77-s + 0.472·79-s − 9.56·83-s + 3.61·85-s + ⋯ |
| L(s) = 1 | + 0.999·5-s + 0.377·7-s − 0.860·11-s − 0.832·13-s + 0.392·17-s − 0.229·19-s + 0.367·23-s + 0.158·29-s − 0.787·31-s + 0.377·35-s − 0.774·37-s + 0.178·41-s − 0.537·43-s + 1.08·47-s + 0.142·49-s − 1.41·53-s − 0.860·55-s + 0.390·59-s − 1.94·61-s − 0.832·65-s + 0.348·67-s + 1.80·71-s + 0.272·73-s − 0.325·77-s + 0.0531·79-s − 1.04·83-s + 0.392·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2.23T + 5T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 23 | \( 1 - 1.76T + 23T^{2} \) |
| 29 | \( 1 - 0.854T + 29T^{2} \) |
| 31 | \( 1 + 4.38T + 31T^{2} \) |
| 37 | \( 1 + 4.70T + 37T^{2} \) |
| 41 | \( 1 - 1.14T + 41T^{2} \) |
| 43 | \( 1 + 3.52T + 43T^{2} \) |
| 47 | \( 1 - 7.47T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 - 2.85T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 2.32T + 73T^{2} \) |
| 79 | \( 1 - 0.472T + 79T^{2} \) |
| 83 | \( 1 + 9.56T + 83T^{2} \) |
| 89 | \( 1 + 5.70T + 89T^{2} \) |
| 97 | \( 1 + 9T + 97T^{2} \) |
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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
−7.43542493027224997737351180844, −6.63673183425818516865636857408, −5.89727521308227951996512601245, −5.23570394305607498854212253347, −4.85098158804577420470472341087, −3.80091515961461830913863109897, −2.83129301551225469370569455608, −2.19402882379646767640728040003, −1.39528374035394068376229403899, 0,
1.39528374035394068376229403899, 2.19402882379646767640728040003, 2.83129301551225469370569455608, 3.80091515961461830913863109897, 4.85098158804577420470472341087, 5.23570394305607498854212253347, 5.89727521308227951996512601245, 6.63673183425818516865636857408, 7.43542493027224997737351180844
