| L(s) = 1 | + 0.751·2-s + 3-s − 1.43·4-s + 3.06·5-s + 0.751·6-s − 2.02·7-s − 2.58·8-s + 9-s + 2.30·10-s − 2.87·11-s − 1.43·12-s − 0.853·13-s − 1.51·14-s + 3.06·15-s + 0.928·16-s − 17-s + 0.751·18-s + 1.01·19-s − 4.39·20-s − 2.02·21-s − 2.16·22-s + 2.89·23-s − 2.58·24-s + 4.36·25-s − 0.641·26-s + 27-s + 2.89·28-s + ⋯ |
| L(s) = 1 | + 0.531·2-s + 0.577·3-s − 0.717·4-s + 1.36·5-s + 0.306·6-s − 0.763·7-s − 0.912·8-s + 0.333·9-s + 0.727·10-s − 0.866·11-s − 0.414·12-s − 0.236·13-s − 0.406·14-s + 0.790·15-s + 0.232·16-s − 0.242·17-s + 0.177·18-s + 0.233·19-s − 0.981·20-s − 0.440·21-s − 0.460·22-s + 0.604·23-s − 0.527·24-s + 0.873·25-s − 0.125·26-s + 0.192·27-s + 0.547·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 0.751T + 2T^{2} \) |
| 5 | \( 1 - 3.06T + 5T^{2} \) |
| 7 | \( 1 + 2.02T + 7T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 13 | \( 1 + 0.853T + 13T^{2} \) |
| 19 | \( 1 - 1.01T + 19T^{2} \) |
| 23 | \( 1 - 2.89T + 23T^{2} \) |
| 29 | \( 1 + 7.11T + 29T^{2} \) |
| 31 | \( 1 - 1.50T + 31T^{2} \) |
| 37 | \( 1 - 8.80T + 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 + 0.143T + 43T^{2} \) |
| 47 | \( 1 + 6.98T + 47T^{2} \) |
| 53 | \( 1 + 7.81T + 53T^{2} \) |
| 59 | \( 1 + 9.82T + 59T^{2} \) |
| 61 | \( 1 + 8.44T + 61T^{2} \) |
| 67 | \( 1 + 0.210T + 67T^{2} \) |
| 71 | \( 1 + 4.66T + 71T^{2} \) |
| 73 | \( 1 - 4.89T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 2.85T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−7.59589227428451644009611084953, −6.50088686213677804988086665856, −6.08076703299366673415518257791, −5.31141097728644968584581208173, −4.79097989552600214402259356437, −3.87027419914777700075283959809, −2.97586600389741487276287735235, −2.56064098297597814187854258236, −1.43736950349119557434044267329, 0,
1.43736950349119557434044267329, 2.56064098297597814187854258236, 2.97586600389741487276287735235, 3.87027419914777700075283959809, 4.79097989552600214402259356437, 5.31141097728644968584581208173, 6.08076703299366673415518257791, 6.50088686213677804988086665856, 7.59589227428451644009611084953
