| L(s) = 1 | − 0.300·2-s + 3-s − 1.90·4-s − 3.45·5-s − 0.300·6-s − 1.15·7-s + 1.17·8-s + 9-s + 1.03·10-s + 6.20·11-s − 1.90·12-s − 0.943·13-s + 0.346·14-s − 3.45·15-s + 3.46·16-s − 7.87·17-s − 0.300·18-s − 8.23·19-s + 6.60·20-s − 1.15·21-s − 1.86·22-s − 23-s + 1.17·24-s + 6.94·25-s + 0.283·26-s + 27-s + 2.20·28-s + ⋯ |
| L(s) = 1 | − 0.212·2-s + 0.577·3-s − 0.954·4-s − 1.54·5-s − 0.122·6-s − 0.435·7-s + 0.414·8-s + 0.333·9-s + 0.328·10-s + 1.87·11-s − 0.551·12-s − 0.261·13-s + 0.0925·14-s − 0.892·15-s + 0.866·16-s − 1.91·17-s − 0.0707·18-s − 1.88·19-s + 1.47·20-s − 0.251·21-s − 0.397·22-s − 0.208·23-s + 0.239·24-s + 1.38·25-s + 0.0555·26-s + 0.192·27-s + 0.416·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8191953198\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8191953198\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.300T + 2T^{2} \) |
| 5 | \( 1 + 3.45T + 5T^{2} \) |
| 7 | \( 1 + 1.15T + 7T^{2} \) |
| 11 | \( 1 - 6.20T + 11T^{2} \) |
| 13 | \( 1 + 0.943T + 13T^{2} \) |
| 17 | \( 1 + 7.87T + 17T^{2} \) |
| 19 | \( 1 + 8.23T + 19T^{2} \) |
| 31 | \( 1 + 3.76T + 31T^{2} \) |
| 37 | \( 1 - 7.78T + 37T^{2} \) |
| 41 | \( 1 + 0.655T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 6.17T + 53T^{2} \) |
| 59 | \( 1 + 2.25T + 59T^{2} \) |
| 61 | \( 1 - 3.80T + 61T^{2} \) |
| 67 | \( 1 - 4.88T + 67T^{2} \) |
| 71 | \( 1 - 2.81T + 71T^{2} \) |
| 73 | \( 1 - 8.34T + 73T^{2} \) |
| 79 | \( 1 - 17.5T + 79T^{2} \) |
| 83 | \( 1 - 4.47T + 83T^{2} \) |
| 89 | \( 1 + 0.0918T + 89T^{2} \) |
| 97 | \( 1 - 2.85T + 97T^{2} \) |
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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
−9.003596117913220226453493413955, −8.579143918020903968503621411703, −7.78297395028064599635779866460, −6.91815083810889948994821607742, −6.25390792897747781878038238231, −4.60293985983852084513376047293, −4.03436899319108704963391461654, −3.79449519442373646318243137869, −2.23892394154124677195651359686, −0.59874909352486061997494031893,
0.59874909352486061997494031893, 2.23892394154124677195651359686, 3.79449519442373646318243137869, 4.03436899319108704963391461654, 4.60293985983852084513376047293, 6.25390792897747781878038238231, 6.91815083810889948994821607742, 7.78297395028064599635779866460, 8.579143918020903968503621411703, 9.003596117913220226453493413955
