| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3672935894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3672935894\) |
| \(L(\frac{5}{2})\) |
|
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
−8.760960515469599586200078467239, −8.164546539859176922532928147151, −7.37142880726088615949587180898, −6.50978874256499355602004341225, −5.60336810403663958770585940321, −4.94281166645378493514087334247, −3.78217404160499395309832210868, −2.73506504140253755132772445379, −2.24711227627512677144299460483, −0.25501605763080462599350759825,
0.25501605763080462599350759825, 2.24711227627512677144299460483, 2.73506504140253755132772445379, 3.78217404160499395309832210868, 4.94281166645378493514087334247, 5.60336810403663958770585940321, 6.50978874256499355602004341225, 7.37142880726088615949587180898, 8.164546539859176922532928147151, 8.760960515469599586200078467239
