| L(s) = 1 | + 3.48·2-s − 2.00·3-s + 4.11·4-s + 5.42·5-s − 6.98·6-s + 27.8·7-s − 13.5·8-s − 22.9·9-s + 18.8·10-s + 11·11-s − 8.25·12-s + 96.8·14-s − 10.8·15-s − 79.9·16-s − 53.2·17-s − 79.9·18-s + 1.47·19-s + 22.2·20-s − 55.8·21-s + 38.2·22-s + 30.0·23-s + 27.1·24-s − 95.6·25-s + 100.·27-s + 114.·28-s − 20.7·29-s − 37.8·30-s + ⋯ |
| L(s) = 1 | + 1.23·2-s − 0.386·3-s + 0.514·4-s + 0.484·5-s − 0.475·6-s + 1.50·7-s − 0.597·8-s − 0.850·9-s + 0.596·10-s + 0.301·11-s − 0.198·12-s + 1.84·14-s − 0.187·15-s − 1.24·16-s − 0.759·17-s − 1.04·18-s + 0.0178·19-s + 0.249·20-s − 0.580·21-s + 0.371·22-s + 0.272·23-s + 0.230·24-s − 0.764·25-s + 0.714·27-s + 0.772·28-s − 0.132·29-s − 0.230·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 3.48T + 8T^{2} \) |
| 3 | \( 1 + 2.00T + 27T^{2} \) |
| 5 | \( 1 - 5.42T + 125T^{2} \) |
| 7 | \( 1 - 27.8T + 343T^{2} \) |
| 17 | \( 1 + 53.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 1.47T + 6.85e3T^{2} \) |
| 23 | \( 1 - 30.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 20.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 15.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 316.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 13.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 244.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 140.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 137.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 160.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 243.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 737.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 331.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 557.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 95.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 86.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + 43.6T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.82e3T + 9.12e5T^{2} \) |
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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
−8.639180832517417079430829406666, −7.57426971422788770869034098884, −6.52581623435589827789440767432, −5.81963739737900042622367446025, −5.14359991693022881655887591824, −4.61530413954533156268679908589, −3.63700840744961450064712013827, −2.51855143375452798161676058720, −1.58705567312661829053821598479, 0,
1.58705567312661829053821598479, 2.51855143375452798161676058720, 3.63700840744961450064712013827, 4.61530413954533156268679908589, 5.14359991693022881655887591824, 5.81963739737900042622367446025, 6.52581623435589827789440767432, 7.57426971422788770869034098884, 8.639180832517417079430829406666
