| L(s) = 1 | − 1.46·2-s + 0.999·3-s − 5.84·4-s + 5·5-s − 1.46·6-s + 27.4·7-s + 20.3·8-s − 26.0·9-s − 7.34·10-s − 11·11-s − 5.84·12-s + 0.949·13-s − 40.3·14-s + 4.99·15-s + 16.9·16-s − 5.50·17-s + 38.1·18-s + 19·19-s − 29.2·20-s + 27.4·21-s + 16.1·22-s + 129.·23-s + 20.3·24-s + 25·25-s − 1.39·26-s − 52.9·27-s − 160.·28-s + ⋯ |
| L(s) = 1 | − 0.519·2-s + 0.192·3-s − 0.730·4-s + 0.447·5-s − 0.0998·6-s + 1.48·7-s + 0.898·8-s − 0.963·9-s − 0.232·10-s − 0.301·11-s − 0.140·12-s + 0.0202·13-s − 0.769·14-s + 0.0860·15-s + 0.264·16-s − 0.0785·17-s + 0.499·18-s + 0.229·19-s − 0.326·20-s + 0.285·21-s + 0.156·22-s + 1.17·23-s + 0.172·24-s + 0.200·25-s − 0.0105·26-s − 0.377·27-s − 1.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.678495471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.678495471\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 1.46T + 8T^{2} \) |
| 3 | \( 1 - 0.999T + 27T^{2} \) |
| 7 | \( 1 - 27.4T + 343T^{2} \) |
| 13 | \( 1 - 0.949T + 2.19e3T^{2} \) |
| 17 | \( 1 + 5.50T + 4.91e3T^{2} \) |
| 23 | \( 1 - 129.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 74.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 38.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 155.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 138.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 377.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 41.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 269.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 492.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 210.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 871.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 512.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 23.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 872.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 36.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.27e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 797.T + 9.12e5T^{2} \) |
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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
−9.188060019138710126156934340501, −8.859164785335837521181493694217, −8.006697228229622107167147550229, −7.43930216872608106516019259810, −5.97904040671868126922403995414, −5.09315309450497872137593973399, −4.52331528965993448117050577365, −3.08542030129842754239360714324, −1.85390943135460035867477160624, −0.76641375006072931906062388316,
0.76641375006072931906062388316, 1.85390943135460035867477160624, 3.08542030129842754239360714324, 4.52331528965993448117050577365, 5.09315309450497872137593973399, 5.97904040671868126922403995414, 7.43930216872608106516019259810, 8.006697228229622107167147550229, 8.859164785335837521181493694217, 9.188060019138710126156934340501
