| L(s) = 1 | + 3·5-s − 7-s − 6·11-s + 2·13-s − 6·17-s + 7·19-s + 3·23-s + 4·25-s − 6·29-s − 2·31-s − 3·35-s + 2·37-s − 2·43-s + 49-s − 6·53-s − 18·55-s + 5·61-s + 6·65-s − 8·67-s + 3·71-s + 2·73-s + 6·77-s − 5·79-s + 12·83-s − 18·85-s − 2·91-s + 21·95-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s − 1.80·11-s + 0.554·13-s − 1.45·17-s + 1.60·19-s + 0.625·23-s + 4/5·25-s − 1.11·29-s − 0.359·31-s − 0.507·35-s + 0.328·37-s − 0.304·43-s + 1/7·49-s − 0.824·53-s − 2.42·55-s + 0.640·61-s + 0.744·65-s − 0.977·67-s + 0.356·71-s + 0.234·73-s + 0.683·77-s − 0.562·79-s + 1.31·83-s − 1.95·85-s − 0.209·91-s + 2.15·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.38476819275535999135405240527, −6.62509384532045389466627231804, −5.94865854409924744021641850334, −5.31974849755602342170659656944, −4.92966912255533521295876588044, −3.73180535825492880643668007815, −2.83036173067638134574426817684, −2.30854670098554420017510854630, −1.36002060033091265869350334541, 0,
1.36002060033091265869350334541, 2.30854670098554420017510854630, 2.83036173067638134574426817684, 3.73180535825492880643668007815, 4.92966912255533521295876588044, 5.31974849755602342170659656944, 5.94865854409924744021641850334, 6.62509384532045389466627231804, 7.38476819275535999135405240527
