| L(s) = 1 | − 2.34·2-s + 2.42·3-s − 2.50·4-s − 18.7·5-s − 5.69·6-s − 0.512·7-s + 24.6·8-s − 21.1·9-s + 43.8·10-s + 5.04·11-s − 6.07·12-s − 48.6·13-s + 1.20·14-s − 45.4·15-s − 37.7·16-s − 6.35·17-s + 49.4·18-s − 123.·19-s + 46.7·20-s − 1.24·21-s − 11.8·22-s + 141.·23-s + 59.8·24-s + 224.·25-s + 114.·26-s − 116.·27-s + 1.28·28-s + ⋯ |
| L(s) = 1 | − 0.829·2-s + 0.467·3-s − 0.312·4-s − 1.67·5-s − 0.387·6-s − 0.0276·7-s + 1.08·8-s − 0.781·9-s + 1.38·10-s + 0.138·11-s − 0.146·12-s − 1.03·13-s + 0.0229·14-s − 0.782·15-s − 0.589·16-s − 0.0907·17-s + 0.647·18-s − 1.49·19-s + 0.523·20-s − 0.0129·21-s − 0.114·22-s + 1.27·23-s + 0.508·24-s + 1.79·25-s + 0.860·26-s − 0.832·27-s + 0.00865·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 2.34T + 8T^{2} \) |
| 3 | \( 1 - 2.42T + 27T^{2} \) |
| 5 | \( 1 + 18.7T + 125T^{2} \) |
| 7 | \( 1 + 0.512T + 343T^{2} \) |
| 11 | \( 1 - 5.04T + 1.33e3T^{2} \) |
| 13 | \( 1 + 48.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.35T + 4.91e3T^{2} \) |
| 19 | \( 1 + 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 141.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 206.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 365.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 175.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 219.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 34.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 771.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 432.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 644.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 919.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 98.7T + 3.89e5T^{2} \) |
| 79 | \( 1 - 411.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 271.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 115.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 623.T + 9.12e5T^{2} \) |
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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
−8.448941602571792483466309999698, −7.980951388856467486356250764998, −7.30707194205630300731220285329, −6.40312291997531260585002257755, −4.80939668453872449982068670141, −4.45359340703033336659572491397, −3.37115338445034228305952352725, −2.45433127301252791915340500962, −0.814517697467941665348562855808, 0,
0.814517697467941665348562855808, 2.45433127301252791915340500962, 3.37115338445034228305952352725, 4.45359340703033336659572491397, 4.80939668453872449982068670141, 6.40312291997531260585002257755, 7.30707194205630300731220285329, 7.980951388856467486356250764998, 8.448941602571792483466309999698
