| L(s) = 1 | + 2.51·2-s + 1.77·3-s − 1.65·4-s + 18.5·5-s + 4.46·6-s − 4.77·7-s − 24.3·8-s − 23.8·9-s + 46.7·10-s − 11·11-s − 2.93·12-s − 12.0·14-s + 32.9·15-s − 48.0·16-s + 71.8·17-s − 60.0·18-s + 28.5·19-s − 30.7·20-s − 8.47·21-s − 27.7·22-s − 217.·23-s − 43.1·24-s + 219.·25-s − 90.2·27-s + 7.91·28-s − 13.6·29-s + 82.9·30-s + ⋯ |
| L(s) = 1 | + 0.890·2-s + 0.341·3-s − 0.206·4-s + 1.65·5-s + 0.304·6-s − 0.258·7-s − 1.07·8-s − 0.883·9-s + 1.47·10-s − 0.301·11-s − 0.0706·12-s − 0.229·14-s + 0.566·15-s − 0.750·16-s + 1.02·17-s − 0.786·18-s + 0.344·19-s − 0.343·20-s − 0.0881·21-s − 0.268·22-s − 1.97·23-s − 0.367·24-s + 1.75·25-s − 0.643·27-s + 0.0534·28-s − 0.0873·29-s + 0.504·30-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 | \( 1 - 2.51T + 8T^{2} \) |
| 3 | \( 1 - 1.77T + 27T^{2} \) |
| 5 | \( 1 - 18.5T + 125T^{2} \) |
| 7 | \( 1 + 4.77T + 343T^{2} \) |
| 17 | \( 1 - 71.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 28.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 217.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 13.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 199.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 277.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 287.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 384.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 39.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 193.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 351.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 741.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 584.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 959.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 935.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 624.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 941.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 257.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 705.T + 9.12e5T^{2} \) |
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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
−8.641959380219319449767886104251, −7.77408445378996988386933306365, −6.43815810625673970303145908701, −5.80179197764121626430579524918, −5.48825866464319727152098928509, −4.44451749080468599508465405499, −3.26620130403256799611773907891, −2.69202037177768898921250194933, −1.61180879180552200489025194508, 0,
1.61180879180552200489025194508, 2.69202037177768898921250194933, 3.26620130403256799611773907891, 4.44451749080468599508465405499, 5.48825866464319727152098928509, 5.80179197764121626430579524918, 6.43815810625673970303145908701, 7.77408445378996988386933306365, 8.641959380219319449767886104251
