| L(s) = 1 | − 1.01·3-s + 0.397·5-s − 14.4·7-s − 25.9·9-s + 52.7·11-s + 60.7·13-s − 0.401·15-s + 0.0555·17-s + 100.·19-s + 14.5·21-s + 15.2·23-s − 124.·25-s + 53.5·27-s + 29·29-s + 172.·31-s − 53.3·33-s − 5.73·35-s + 305.·37-s − 61.4·39-s + 318.·41-s + 467.·43-s − 10.3·45-s + 249.·47-s − 134.·49-s − 0.0561·51-s − 201.·53-s + 20.9·55-s + ⋯ |
| L(s) = 1 | − 0.194·3-s + 0.0355·5-s − 0.778·7-s − 0.962·9-s + 1.44·11-s + 1.29·13-s − 0.00691·15-s + 0.000792·17-s + 1.20·19-s + 0.151·21-s + 0.138·23-s − 0.998·25-s + 0.381·27-s + 0.185·29-s + 0.998·31-s − 0.281·33-s − 0.0276·35-s + 1.35·37-s − 0.252·39-s + 1.21·41-s + 1.65·43-s − 0.0341·45-s + 0.774·47-s − 0.393·49-s − 0.000154·51-s − 0.522·53-s + 0.0514·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.610395912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610395912\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 1.01T + 27T^{2} \) |
| 5 | \( 1 - 0.397T + 125T^{2} \) |
| 7 | \( 1 + 14.4T + 343T^{2} \) |
| 11 | \( 1 - 52.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 60.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 0.0555T + 4.91e3T^{2} \) |
| 19 | \( 1 - 100.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 15.2T + 1.21e4T^{2} \) |
| 31 | \( 1 - 172.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 305.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 318.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 467.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 249.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 201.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 696.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 796.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 828.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 676.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 735.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 149.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 947.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 80.8T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.03e3T + 9.12e5T^{2} \) |
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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
−11.64696139983853770972733723491, −11.00956692128402544404785300720, −9.599881372734780589320538990339, −9.011092378049379371045698439593, −7.78784995903535119486034839695, −6.35667756737013590541255639041, −5.88119402937595363399384071081, −4.12758115959821017742016597078, −3.01106774249532677464975113270, −0.994087863048673928054519246561,
0.994087863048673928054519246561, 3.01106774249532677464975113270, 4.12758115959821017742016597078, 5.88119402937595363399384071081, 6.35667756737013590541255639041, 7.78784995903535119486034839695, 9.011092378049379371045698439593, 9.599881372734780589320538990339, 11.00956692128402544404785300720, 11.64696139983853770972733723491
