| L(s) = 1 | + 1.56·2-s − 3-s + 0.449·4-s + 1.70·5-s − 1.56·6-s + 7-s − 2.42·8-s + 9-s + 2.67·10-s − 4.11·11-s − 0.449·12-s + 1.47·13-s + 1.56·14-s − 1.70·15-s − 4.69·16-s + 2.15·17-s + 1.56·18-s + 6.20·19-s + 0.767·20-s − 21-s − 6.43·22-s − 7.80·23-s + 2.42·24-s − 2.08·25-s + 2.30·26-s − 27-s + 0.449·28-s + ⋯ |
| L(s) = 1 | + 1.10·2-s − 0.577·3-s + 0.224·4-s + 0.763·5-s − 0.638·6-s + 0.377·7-s − 0.857·8-s + 0.333·9-s + 0.844·10-s − 1.23·11-s − 0.129·12-s + 0.409·13-s + 0.418·14-s − 0.440·15-s − 1.17·16-s + 0.522·17-s + 0.368·18-s + 1.42·19-s + 0.171·20-s − 0.218·21-s − 1.37·22-s − 1.62·23-s + 0.495·24-s − 0.417·25-s + 0.452·26-s − 0.192·27-s + 0.0849·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4767 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4767 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.843332983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.843332983\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 227 | \( 1 + T \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 5 | \( 1 - 1.70T + 5T^{2} \) |
| 11 | \( 1 + 4.11T + 11T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 - 2.15T + 17T^{2} \) |
| 19 | \( 1 - 6.20T + 19T^{2} \) |
| 23 | \( 1 + 7.80T + 23T^{2} \) |
| 29 | \( 1 - 0.926T + 29T^{2} \) |
| 31 | \( 1 - 8.21T + 31T^{2} \) |
| 37 | \( 1 - 5.77T + 37T^{2} \) |
| 41 | \( 1 - 8.51T + 41T^{2} \) |
| 43 | \( 1 - 2.17T + 43T^{2} \) |
| 47 | \( 1 - 2.18T + 47T^{2} \) |
| 53 | \( 1 - 0.0760T + 53T^{2} \) |
| 59 | \( 1 + 5.37T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 + 6.41T + 73T^{2} \) |
| 79 | \( 1 - 5.32T + 79T^{2} \) |
| 83 | \( 1 - 3.14T + 83T^{2} \) |
| 89 | \( 1 - 1.32T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.040355385800618145252647652055, −7.59055070655950792347186860188, −6.38490676705549980609651172588, −5.81976337279749825617666853351, −5.45697951237960566557470351244, −4.70436019138149566353805489867, −3.97540029307712197019454050974, −2.95970294082877392047544305406, −2.18050071907658939562947155633, −0.812090276789535887855119779095,
0.812090276789535887855119779095, 2.18050071907658939562947155633, 2.95970294082877392047544305406, 3.97540029307712197019454050974, 4.70436019138149566353805489867, 5.45697951237960566557470351244, 5.81976337279749825617666853351, 6.38490676705549980609651172588, 7.59055070655950792347186860188, 8.040355385800618145252647652055
