| L(s) = 1 | + 1.74·2-s + 3-s + 1.04·4-s + 0.892·5-s + 1.74·6-s + 3.89·7-s − 1.66·8-s + 9-s + 1.55·10-s + 4.47·11-s + 1.04·12-s + 2.21·13-s + 6.79·14-s + 0.892·15-s − 4.99·16-s − 5.25·17-s + 1.74·18-s + 0.817·19-s + 0.931·20-s + 3.89·21-s + 7.80·22-s + 23-s − 1.66·24-s − 4.20·25-s + 3.86·26-s + 27-s + 4.06·28-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 0.577·3-s + 0.521·4-s + 0.399·5-s + 0.712·6-s + 1.47·7-s − 0.589·8-s + 0.333·9-s + 0.492·10-s + 1.34·11-s + 0.301·12-s + 0.613·13-s + 1.81·14-s + 0.230·15-s − 1.24·16-s − 1.27·17-s + 0.411·18-s + 0.187·19-s + 0.208·20-s + 0.850·21-s + 1.66·22-s + 0.208·23-s − 0.340·24-s − 0.840·25-s + 0.757·26-s + 0.192·27-s + 0.768·28-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
| \(L(1)\) |
\(\approx\) |
\(5.007237878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.007237878\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.74T + 2T^{2} \) |
| 5 | \( 1 - 0.892T + 5T^{2} \) |
| 7 | \( 1 - 3.89T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 - 2.21T + 13T^{2} \) |
| 17 | \( 1 + 5.25T + 17T^{2} \) |
| 19 | \( 1 - 0.817T + 19T^{2} \) |
| 31 | \( 1 + 4.19T + 31T^{2} \) |
| 37 | \( 1 - 8.35T + 37T^{2} \) |
| 41 | \( 1 + 1.95T + 41T^{2} \) |
| 43 | \( 1 + 1.55T + 43T^{2} \) |
| 47 | \( 1 + 1.09T + 47T^{2} \) |
| 53 | \( 1 + 0.517T + 53T^{2} \) |
| 59 | \( 1 + 6.99T + 59T^{2} \) |
| 61 | \( 1 + 0.394T + 61T^{2} \) |
| 67 | \( 1 - 7.01T + 67T^{2} \) |
| 71 | \( 1 + 4.42T + 71T^{2} \) |
| 73 | \( 1 + 0.726T + 73T^{2} \) |
| 79 | \( 1 + 7.50T + 79T^{2} \) |
| 83 | \( 1 - 8.09T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.983756452475177021373177034806, −8.535269142615984970311249405840, −7.52001712699328972762075426655, −6.53974701077949128641679923005, −5.89352845677617556211688528059, −4.87312599580242197056739888874, −4.28378006039054869652154773928, −3.58931489384576846967505480549, −2.34441145968962614936021663926, −1.47320495248686258520529750090,
1.47320495248686258520529750090, 2.34441145968962614936021663926, 3.58931489384576846967505480549, 4.28378006039054869652154773928, 4.87312599580242197056739888874, 5.89352845677617556211688528059, 6.53974701077949128641679923005, 7.52001712699328972762075426655, 8.535269142615984970311249405840, 8.983756452475177021373177034806
