| L(s) = 1 | + 1.63·2-s − 2.06·3-s − 5.32·4-s + 14.3·5-s − 3.37·6-s + 27.7·7-s − 21.8·8-s − 22.7·9-s + 23.5·10-s + 11·11-s + 10.9·12-s + 45.4·14-s − 29.6·15-s + 6.89·16-s − 44.5·17-s − 37.2·18-s + 53.2·19-s − 76.4·20-s − 57.3·21-s + 18.0·22-s − 100.·23-s + 45.0·24-s + 81.5·25-s + 102.·27-s − 147.·28-s − 269.·29-s − 48.5·30-s + ⋯ |
| L(s) = 1 | + 0.578·2-s − 0.397·3-s − 0.665·4-s + 1.28·5-s − 0.229·6-s + 1.49·7-s − 0.963·8-s − 0.842·9-s + 0.743·10-s + 0.301·11-s + 0.264·12-s + 0.867·14-s − 0.510·15-s + 0.107·16-s − 0.635·17-s − 0.487·18-s + 0.643·19-s − 0.855·20-s − 0.595·21-s + 0.174·22-s − 0.912·23-s + 0.382·24-s + 0.652·25-s + 0.731·27-s − 0.997·28-s − 1.72·29-s − 0.295·30-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 | \( 1 - 1.63T + 8T^{2} \) |
| 3 | \( 1 + 2.06T + 27T^{2} \) |
| 5 | \( 1 - 14.3T + 125T^{2} \) |
| 7 | \( 1 - 27.7T + 343T^{2} \) |
| 17 | \( 1 + 44.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 53.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 100.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 269.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 202.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 360.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 174.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 342.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 594.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 648.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 217.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 730.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 233.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 183.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 547.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.16e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.74e3T + 9.12e5T^{2} \) |
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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
−8.575267491397679238525807948601, −7.83479105186614158957905901977, −6.57367491202160774652013633692, −5.67526583643595704334151347093, −5.38324645260523087394092035992, −4.61650120535586757530543941615, −3.59089943337786395224971164715, −2.30154875470476188906916022102, −1.45213101101582063274187486472, 0,
1.45213101101582063274187486472, 2.30154875470476188906916022102, 3.59089943337786395224971164715, 4.61650120535586757530543941615, 5.38324645260523087394092035992, 5.67526583643595704334151347093, 6.57367491202160774652013633692, 7.83479105186614158957905901977, 8.575267491397679238525807948601
