| L(s) = 1 | + 1.79·2-s − 2.53·3-s + 1.23·4-s − 5-s − 4.56·6-s + 0.104·7-s − 1.36·8-s + 3.43·9-s − 1.79·10-s − 11-s − 3.14·12-s + 5.51·13-s + 0.187·14-s + 2.53·15-s − 4.94·16-s − 1.18·17-s + 6.18·18-s + 19-s − 1.23·20-s − 0.263·21-s − 1.79·22-s + 8.21·23-s + 3.47·24-s + 25-s + 9.92·26-s − 1.11·27-s + 0.128·28-s + ⋯ |
| L(s) = 1 | + 1.27·2-s − 1.46·3-s + 0.619·4-s − 0.447·5-s − 1.86·6-s + 0.0393·7-s − 0.484·8-s + 1.14·9-s − 0.569·10-s − 0.301·11-s − 0.907·12-s + 1.52·13-s + 0.0500·14-s + 0.655·15-s − 1.23·16-s − 0.288·17-s + 1.45·18-s + 0.229·19-s − 0.277·20-s − 0.0575·21-s − 0.383·22-s + 1.71·23-s + 0.709·24-s + 0.200·25-s + 1.94·26-s − 0.214·27-s + 0.0243·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.618277520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.618277520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.79T + 2T^{2} \) |
| 3 | \( 1 + 2.53T + 3T^{2} \) |
| 7 | \( 1 - 0.104T + 7T^{2} \) |
| 13 | \( 1 - 5.51T + 13T^{2} \) |
| 17 | \( 1 + 1.18T + 17T^{2} \) |
| 23 | \( 1 - 8.21T + 23T^{2} \) |
| 29 | \( 1 - 7.19T + 29T^{2} \) |
| 31 | \( 1 - 6.46T + 31T^{2} \) |
| 37 | \( 1 + 5.45T + 37T^{2} \) |
| 41 | \( 1 + 3.05T + 41T^{2} \) |
| 43 | \( 1 - 6.82T + 43T^{2} \) |
| 47 | \( 1 - 1.45T + 47T^{2} \) |
| 53 | \( 1 + 7.94T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 5.96T + 61T^{2} \) |
| 67 | \( 1 + 1.57T + 67T^{2} \) |
| 71 | \( 1 - 15.9T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 2.66T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−10.39884935601776834115139967597, −9.066733912037897682184424429140, −8.198312353134374520519578630874, −6.70937851488929644647862071494, −6.45742845932582932330188438685, −5.37575928678542678177663683384, −4.89949149553985028748280597337, −3.96824682834420163058620808789, −2.95379176933922193558098644239, −0.895395658803804308921796376434,
0.895395658803804308921796376434, 2.95379176933922193558098644239, 3.96824682834420163058620808789, 4.89949149553985028748280597337, 5.37575928678542678177663683384, 6.45742845932582932330188438685, 6.70937851488929644647862071494, 8.198312353134374520519578630874, 9.066733912037897682184424429140, 10.39884935601776834115139967597
