| L(s) = 1 | + 0.792·2-s − 0.186·3-s − 1.37·4-s + 0.0889·5-s − 0.147·6-s − 7-s − 2.67·8-s − 2.96·9-s + 0.0705·10-s + 3.65·11-s + 0.255·12-s + 3.57·13-s − 0.792·14-s − 0.0166·15-s + 0.626·16-s + 2.55·17-s − 2.34·18-s − 3.97·19-s − 0.122·20-s + 0.186·21-s + 2.89·22-s − 0.443·23-s + 0.498·24-s − 4.99·25-s + 2.83·26-s + 1.11·27-s + 1.37·28-s + ⋯ |
| L(s) = 1 | + 0.560·2-s − 0.107·3-s − 0.685·4-s + 0.0398·5-s − 0.0603·6-s − 0.377·7-s − 0.944·8-s − 0.988·9-s + 0.0223·10-s + 1.10·11-s + 0.0738·12-s + 0.990·13-s − 0.211·14-s − 0.00428·15-s + 0.156·16-s + 0.620·17-s − 0.553·18-s − 0.911·19-s − 0.0273·20-s + 0.0407·21-s + 0.616·22-s − 0.0925·23-s + 0.101·24-s − 0.998·25-s + 0.555·26-s + 0.214·27-s + 0.259·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1589 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1589 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.586829570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.586829570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 227 | \( 1 - T \) |
| good | 2 | \( 1 - 0.792T + 2T^{2} \) |
| 3 | \( 1 + 0.186T + 3T^{2} \) |
| 5 | \( 1 - 0.0889T + 5T^{2} \) |
| 11 | \( 1 - 3.65T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 - 2.55T + 17T^{2} \) |
| 19 | \( 1 + 3.97T + 19T^{2} \) |
| 23 | \( 1 + 0.443T + 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 - 4.88T + 31T^{2} \) |
| 37 | \( 1 - 0.524T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 2.71T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 5.02T + 59T^{2} \) |
| 61 | \( 1 - 6.46T + 61T^{2} \) |
| 67 | \( 1 - 5.73T + 67T^{2} \) |
| 71 | \( 1 + 3.96T + 71T^{2} \) |
| 73 | \( 1 - 3.44T + 73T^{2} \) |
| 79 | \( 1 - 5.78T + 79T^{2} \) |
| 83 | \( 1 - 0.896T + 83T^{2} \) |
| 89 | \( 1 - 2.96T + 89T^{2} \) |
| 97 | \( 1 + 4.10T + 97T^{2} \) |
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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
−9.276818887162262982336085896066, −8.692440852510487972067536513702, −8.053236954531462376515619448634, −6.65931483835220908844850896911, −6.01419066413532494735053053746, −5.44374044247819433427048233432, −4.15647088309756671474307666757, −3.72094022305370013657665992058, −2.58045216286733143021047792430, −0.829371001484912887722726326769,
0.829371001484912887722726326769, 2.58045216286733143021047792430, 3.72094022305370013657665992058, 4.15647088309756671474307666757, 5.44374044247819433427048233432, 6.01419066413532494735053053746, 6.65931483835220908844850896911, 8.053236954531462376515619448634, 8.692440852510487972067536513702, 9.276818887162262982336085896066
