| L(s) = 1 | − 2-s − 2·3-s − 4-s + 5-s + 2·6-s − 2·7-s + 3·8-s + 9-s − 10-s + 2·12-s + 13-s + 2·14-s − 2·15-s − 16-s + 5·17-s − 18-s − 6·19-s − 20-s + 4·21-s + 2·23-s − 6·24-s − 4·25-s − 26-s + 4·27-s + 2·28-s + 2·30-s + 2·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.447·5-s + 0.816·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s + 0.277·13-s + 0.534·14-s − 0.516·15-s − 1/4·16-s + 1.21·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s + 0.872·21-s + 0.417·23-s − 1.22·24-s − 4/5·25-s − 0.196·26-s + 0.769·27-s + 0.377·28-s + 0.365·30-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101761 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101761 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9281927183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9281927183\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.70557752840090, −12.99303310768875, −12.88514219363888, −12.25607794464601, −11.74742592149829, −11.14733210012567, −10.69251676304895, −10.20821929384465, −9.932799451224890, −9.321617747543549, −8.934921129787278, −8.225931602375893, −7.888560232999870, −7.142901497493628, −6.580607685724356, −6.054634838358173, −5.747756565123244, −5.057047304849804, −4.627462389906404, −3.868219076586001, −3.382814133675957, −2.434451192094781, −1.766639281541389, −0.7781593229434257, −0.5563067698191174,
0.5563067698191174, 0.7781593229434257, 1.766639281541389, 2.434451192094781, 3.382814133675957, 3.868219076586001, 4.627462389906404, 5.057047304849804, 5.747756565123244, 6.054634838358173, 6.580607685724356, 7.142901497493628, 7.888560232999870, 8.225931602375893, 8.934921129787278, 9.321617747543549, 9.932799451224890, 10.20821929384465, 10.69251676304895, 11.14733210012567, 11.74742592149829, 12.25607794464601, 12.88514219363888, 12.99303310768875, 13.70557752840090
