| L(s) = 1 | − 3-s + 3·5-s + 9-s + 4·11-s − 3·13-s − 3·15-s + 4·17-s − 23-s + 4·25-s − 27-s − 3·29-s − 6·31-s − 4·33-s + 9·37-s + 3·39-s − 9·41-s − 3·43-s + 3·45-s − 7·47-s − 4·51-s + 4·53-s + 12·55-s − 6·59-s + 10·61-s − 9·65-s + 4·67-s + 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1/3·9-s + 1.20·11-s − 0.832·13-s − 0.774·15-s + 0.970·17-s − 0.208·23-s + 4/5·25-s − 0.192·27-s − 0.557·29-s − 1.07·31-s − 0.696·33-s + 1.47·37-s + 0.480·39-s − 1.40·41-s − 0.457·43-s + 0.447·45-s − 1.02·47-s − 0.560·51-s + 0.549·53-s + 1.61·55-s − 0.781·59-s + 1.28·61-s − 1.11·65-s + 0.488·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.12091424060902, −12.77446037326404, −12.33562313148825, −11.72991725240107, −11.46836191778377, −10.93739290881852, −10.20827649408135, −9.908309658312781, −9.651721288533216, −9.143133198810832, −8.655627947206124, −7.915301673615536, −7.454919940694514, −6.861148858693758, −6.413222862949279, −6.068650304411520, −5.413822333426500, −5.180828099318172, −4.578819955875465, −3.789693883394012, −3.453857067177777, −2.563127544474797, −2.013934427662691, −1.502612890333851, −0.9414431450082615, 0,
0.9414431450082615, 1.502612890333851, 2.013934427662691, 2.563127544474797, 3.453857067177777, 3.789693883394012, 4.578819955875465, 5.180828099318172, 5.413822333426500, 6.068650304411520, 6.413222862949279, 6.861148858693758, 7.454919940694514, 7.915301673615536, 8.655627947206124, 9.143133198810832, 9.651721288533216, 9.908309658312781, 10.20827649408135, 10.93739290881852, 11.46836191778377, 11.72991725240107, 12.33562313148825, 12.77446037326404, 13.12091424060902
