| L(s) = 1 | − 3-s − 5-s + 9-s + 4·11-s + 13-s + 15-s − 4·17-s + 2·23-s + 25-s − 27-s + 29-s + 4·31-s − 4·33-s − 2·37-s − 39-s − 4·43-s − 45-s − 7·49-s + 4·51-s + 2·53-s − 4·55-s + 2·59-s − 10·61-s − 65-s − 4·67-s − 2·69-s − 2·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.258·15-s − 0.970·17-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 0.185·29-s + 0.718·31-s − 0.696·33-s − 0.328·37-s − 0.160·39-s − 0.609·43-s − 0.149·45-s − 49-s + 0.560·51-s + 0.274·53-s − 0.539·55-s + 0.260·59-s − 1.28·61-s − 0.124·65-s − 0.488·67-s − 0.240·69-s − 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.58223108396049, −12.20771529078316, −11.82030749967936, −11.44120303169811, −10.96706839353451, −10.65105284698075, −10.10891827632466, −9.398945498312267, −9.250305671973740, −8.678264868350951, −8.084165191142154, −7.842625966518646, −6.925451772031685, −6.630620483940615, −6.573946140262414, −5.700309622426017, −5.332000324353969, −4.585823258635518, −4.332237348449806, −3.840740887147163, −3.197586327200930, −2.702888227227743, −1.815259221595797, −1.401788066652918, −0.7113060595276051, 0,
0.7113060595276051, 1.401788066652918, 1.815259221595797, 2.702888227227743, 3.197586327200930, 3.840740887147163, 4.332237348449806, 4.585823258635518, 5.332000324353969, 5.700309622426017, 6.573946140262414, 6.630620483940615, 6.925451772031685, 7.842625966518646, 8.084165191142154, 8.678264868350951, 9.250305671973740, 9.398945498312267, 10.10891827632466, 10.65105284698075, 10.96706839353451, 11.44120303169811, 11.82030749967936, 12.20771529078316, 12.58223108396049
