| L(s) = 1 | − 1.35·2-s − 2.82·3-s − 0.167·4-s + 0.473·5-s + 3.82·6-s + 7-s + 2.93·8-s + 4.99·9-s − 0.641·10-s − 11-s + 0.473·12-s − 13-s − 1.35·14-s − 1.34·15-s − 3.63·16-s − 1.18·17-s − 6.76·18-s + 2.64·19-s − 0.0794·20-s − 2.82·21-s + 1.35·22-s + 0.986·23-s − 8.29·24-s − 4.77·25-s + 1.35·26-s − 5.64·27-s − 0.167·28-s + ⋯ |
| L(s) = 1 | − 0.957·2-s − 1.63·3-s − 0.0838·4-s + 0.211·5-s + 1.56·6-s + 0.377·7-s + 1.03·8-s + 1.66·9-s − 0.202·10-s − 0.301·11-s + 0.136·12-s − 0.277·13-s − 0.361·14-s − 0.345·15-s − 0.909·16-s − 0.286·17-s − 1.59·18-s + 0.606·19-s − 0.0177·20-s − 0.617·21-s + 0.288·22-s + 0.205·23-s − 1.69·24-s − 0.955·25-s + 0.265·26-s − 1.08·27-s − 0.0316·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4032736052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4032736052\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 0.473T + 5T^{2} \) |
| 17 | \( 1 + 1.18T + 17T^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 23 | \( 1 - 0.986T + 23T^{2} \) |
| 29 | \( 1 + 5.28T + 29T^{2} \) |
| 31 | \( 1 - 0.0521T + 31T^{2} \) |
| 37 | \( 1 + 3.53T + 37T^{2} \) |
| 41 | \( 1 + 0.353T + 41T^{2} \) |
| 43 | \( 1 - 0.0473T + 43T^{2} \) |
| 47 | \( 1 - 2.98T + 47T^{2} \) |
| 53 | \( 1 + 6.84T + 53T^{2} \) |
| 59 | \( 1 - 7.94T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 8.99T + 67T^{2} \) |
| 71 | \( 1 - 9.31T + 71T^{2} \) |
| 73 | \( 1 + 1.11T + 73T^{2} \) |
| 79 | \( 1 + 0.181T + 79T^{2} \) |
| 83 | \( 1 - 1.48T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 + 5.00T + 97T^{2} \) |
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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
−9.970115597779079513202603240155, −9.419876211145011839127308502397, −8.301155165617694081897401529300, −7.45356254626022438183549175852, −6.69162171767787445390324627410, −5.53085123654131352949144828790, −5.05242752356995650772065130454, −3.99020954848054038023803854705, −1.90428416060025497711539930512, −0.62637405028887366928638761304,
0.62637405028887366928638761304, 1.90428416060025497711539930512, 3.99020954848054038023803854705, 5.05242752356995650772065130454, 5.53085123654131352949144828790, 6.69162171767787445390324627410, 7.45356254626022438183549175852, 8.301155165617694081897401529300, 9.419876211145011839127308502397, 9.970115597779079513202603240155
