| L(s) = 1 | + 3-s + 9-s − 2·17-s + 19-s − 6·23-s − 5·25-s + 27-s − 10·29-s − 10·31-s − 8·37-s + 6·41-s − 4·43-s + 6·47-s − 7·49-s − 2·51-s + 6·53-s + 57-s + 12·59-s + 10·61-s + 8·67-s − 6·69-s − 8·71-s − 2·73-s − 5·75-s − 6·79-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.485·17-s + 0.229·19-s − 1.25·23-s − 25-s + 0.192·27-s − 1.85·29-s − 1.79·31-s − 1.31·37-s + 0.937·41-s − 0.609·43-s + 0.875·47-s − 49-s − 0.280·51-s + 0.824·53-s + 0.132·57-s + 1.56·59-s + 1.28·61-s + 0.977·67-s − 0.722·69-s − 0.949·71-s − 0.234·73-s − 0.577·75-s − 0.675·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.168970345890267096694431770502, −7.46673926194569887500692717232, −6.88565694981038390632634000087, −5.80792412759310459185090671492, −5.27011383596716056041376055900, −3.93470913415548969633528171479, −3.72465779285701629479354216953, −2.36515606401106323529575170328, −1.71729500698904191815053448386, 0,
1.71729500698904191815053448386, 2.36515606401106323529575170328, 3.72465779285701629479354216953, 3.93470913415548969633528171479, 5.27011383596716056041376055900, 5.80792412759310459185090671492, 6.88565694981038390632634000087, 7.46673926194569887500692717232, 8.168970345890267096694431770502
