| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 3·5-s − 2·6-s + 5·7-s + 9-s − 6·10-s − 2·12-s − 2·13-s + 10·14-s + 3·15-s − 4·16-s + 17-s + 2·18-s + 19-s − 6·20-s − 5·21-s − 4·23-s + 4·25-s − 4·26-s − 27-s + 10·28-s + 2·29-s + 6·30-s − 6·31-s − 8·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 1.34·5-s − 0.816·6-s + 1.88·7-s + 1/3·9-s − 1.89·10-s − 0.577·12-s − 0.554·13-s + 2.67·14-s + 0.774·15-s − 16-s + 0.242·17-s + 0.471·18-s + 0.229·19-s − 1.34·20-s − 1.09·21-s − 0.834·23-s + 4/5·25-s − 0.784·26-s − 0.192·27-s + 1.88·28-s + 0.371·29-s + 1.09·30-s − 1.07·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.43755268339255182270866775688, −6.95333653112949028642183365593, −5.82201250996795631397735521238, −5.30575361876571555675709307141, −4.64831433194343792963963650578, −4.23023339125662356328251569037, −3.57553622245677274525627107284, −2.47849673392805399692938216213, −1.44645188395830487846696686994, 0,
1.44645188395830487846696686994, 2.47849673392805399692938216213, 3.57553622245677274525627107284, 4.23023339125662356328251569037, 4.64831433194343792963963650578, 5.30575361876571555675709307141, 5.82201250996795631397735521238, 6.95333653112949028642183365593, 7.43755268339255182270866775688
