| L(s) = 1 | − 3-s − 2·5-s + 2·7-s + 9-s − 2·13-s + 2·15-s − 4·17-s − 6·19-s − 2·21-s − 25-s − 27-s − 8·29-s − 8·31-s − 4·35-s − 10·37-s + 2·39-s − 8·41-s − 2·43-s − 2·45-s − 8·47-s − 3·49-s + 4·51-s + 2·53-s + 6·57-s − 12·59-s + 10·61-s + 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s − 0.970·17-s − 1.37·19-s − 0.436·21-s − 1/5·25-s − 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.676·35-s − 1.64·37-s + 0.320·39-s − 1.24·41-s − 0.304·43-s − 0.298·45-s − 1.16·47-s − 3/7·49-s + 0.560·51-s + 0.274·53-s + 0.794·57-s − 1.56·59-s + 1.28·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−16.07151343376059, −15.31925932656467, −14.93715413369038, −14.73202675395121, −13.75243125781431, −13.27445027919837, −12.60203664300314, −12.20142037339500, −11.57023192490572, −11.09063938409442, −10.81678409657385, −10.10109620827608, −9.342000473839292, −8.721712191905006, −8.191645328076428, −7.609886518008811, −6.987916175323969, −6.560463682239176, −5.639636753851219, −5.086763598563966, −4.494675021618772, −3.924558228805985, −3.274347855143445, −1.986374781128185, −1.728135697432896, 0, 0,
1.728135697432896, 1.986374781128185, 3.274347855143445, 3.924558228805985, 4.494675021618772, 5.086763598563966, 5.639636753851219, 6.560463682239176, 6.987916175323969, 7.609886518008811, 8.191645328076428, 8.721712191905006, 9.342000473839292, 10.10109620827608, 10.81678409657385, 11.09063938409442, 11.57023192490572, 12.20142037339500, 12.60203664300314, 13.27445027919837, 13.75243125781431, 14.73202675395121, 14.93715413369038, 15.31925932656467, 16.07151343376059
