| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 3·5-s − 2·6-s − 7-s + 9-s + 6·10-s + 2·12-s + 2·14-s − 3·15-s − 4·16-s + 2·17-s − 2·18-s − 19-s − 6·20-s − 21-s + 23-s + 4·25-s + 27-s − 2·28-s + 5·29-s + 6·30-s − 5·31-s + 8·32-s − 4·34-s + 3·35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 1.34·5-s − 0.816·6-s − 0.377·7-s + 1/3·9-s + 1.89·10-s + 0.577·12-s + 0.534·14-s − 0.774·15-s − 16-s + 0.485·17-s − 0.471·18-s − 0.229·19-s − 1.34·20-s − 0.218·21-s + 0.208·23-s + 4/5·25-s + 0.192·27-s − 0.377·28-s + 0.928·29-s + 1.09·30-s − 0.898·31-s + 1.41·32-s − 0.685·34-s + 0.507·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−8.291337964577720542000036601017, −7.59632976338319477439809420770, −7.23113173664801018219950626461, −6.36768099124137407662455495596, −5.04305933288398388366326087917, −4.09634718745809111617687686594, −3.40040765767289635727722931599, −2.35584928239184963657299205645, −1.11612014257101683402638604868, 0,
1.11612014257101683402638604868, 2.35584928239184963657299205645, 3.40040765767289635727722931599, 4.09634718745809111617687686594, 5.04305933288398388366326087917, 6.36768099124137407662455495596, 7.23113173664801018219950626461, 7.59632976338319477439809420770, 8.291337964577720542000036601017
