| L(s) = 1 | − 2.63·2-s + 3-s + 4.93·4-s + 0.107·5-s − 2.63·6-s + 4.37·7-s − 7.72·8-s + 9-s − 0.281·10-s + 0.577·11-s + 4.93·12-s − 4.16·13-s − 11.5·14-s + 0.107·15-s + 10.4·16-s − 3.07·17-s − 2.63·18-s + 3.22·19-s + 0.528·20-s + 4.37·21-s − 1.52·22-s − 23-s − 7.72·24-s − 4.98·25-s + 10.9·26-s + 27-s + 21.5·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 0.577·3-s + 2.46·4-s + 0.0478·5-s − 1.07·6-s + 1.65·7-s − 2.73·8-s + 0.333·9-s − 0.0891·10-s + 0.174·11-s + 1.42·12-s − 1.15·13-s − 3.07·14-s + 0.0276·15-s + 2.61·16-s − 0.746·17-s − 0.620·18-s + 0.740·19-s + 0.118·20-s + 0.954·21-s − 0.324·22-s − 0.208·23-s − 1.57·24-s − 0.997·25-s + 2.15·26-s + 0.192·27-s + 4.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.109686679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.109686679\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 5 | \( 1 - 0.107T + 5T^{2} \) |
| 7 | \( 1 - 4.37T + 7T^{2} \) |
| 11 | \( 1 - 0.577T + 11T^{2} \) |
| 13 | \( 1 + 4.16T + 13T^{2} \) |
| 17 | \( 1 + 3.07T + 17T^{2} \) |
| 19 | \( 1 - 3.22T + 19T^{2} \) |
| 31 | \( 1 - 6.08T + 31T^{2} \) |
| 37 | \( 1 - 4.83T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 0.619T + 43T^{2} \) |
| 47 | \( 1 - 0.421T + 47T^{2} \) |
| 53 | \( 1 - 2.63T + 53T^{2} \) |
| 59 | \( 1 + 0.709T + 59T^{2} \) |
| 61 | \( 1 - 8.89T + 61T^{2} \) |
| 67 | \( 1 - 2.95T + 67T^{2} \) |
| 71 | \( 1 - 2.68T + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 - 0.714T + 79T^{2} \) |
| 83 | \( 1 - 4.88T + 83T^{2} \) |
| 89 | \( 1 - 9.26T + 89T^{2} \) |
| 97 | \( 1 + 5.58T + 97T^{2} \) |
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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
−9.175046435665678588856967675195, −8.322369862230286574118864219642, −7.79590580832437351665176698120, −7.39617056457485883308894310600, −6.40162839699108277459474336729, −5.22963050976393439129998736694, −4.17397911223651482110810972415, −2.56235192795104578922438497148, −2.03185869378448859044974400373, −0.926894161411607642938072113413,
0.926894161411607642938072113413, 2.03185869378448859044974400373, 2.56235192795104578922438497148, 4.17397911223651482110810972415, 5.22963050976393439129998736694, 6.40162839699108277459474336729, 7.39617056457485883308894310600, 7.79590580832437351665176698120, 8.322369862230286574118864219642, 9.175046435665678588856967675195
