| L(s) = 1 | + 2-s − 0.453·3-s + 4-s − 0.630·5-s − 0.453·6-s + 1.98·7-s + 8-s − 2.79·9-s − 0.630·10-s − 5.29·11-s − 0.453·12-s − 0.116·13-s + 1.98·14-s + 0.285·15-s + 16-s + 7.82·17-s − 2.79·18-s − 4.20·19-s − 0.630·20-s − 0.902·21-s − 5.29·22-s + 0.805·23-s − 0.453·24-s − 4.60·25-s − 0.116·26-s + 2.62·27-s + 1.98·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.262·3-s + 0.5·4-s − 0.281·5-s − 0.185·6-s + 0.751·7-s + 0.353·8-s − 0.931·9-s − 0.199·10-s − 1.59·11-s − 0.131·12-s − 0.0323·13-s + 0.531·14-s + 0.0738·15-s + 0.250·16-s + 1.89·17-s − 0.658·18-s − 0.964·19-s − 0.140·20-s − 0.196·21-s − 1.12·22-s + 0.167·23-s − 0.0926·24-s − 0.920·25-s − 0.0229·26-s + 0.506·27-s + 0.375·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 + 0.453T + 3T^{2} \) |
| 5 | \( 1 + 0.630T + 5T^{2} \) |
| 7 | \( 1 - 1.98T + 7T^{2} \) |
| 11 | \( 1 + 5.29T + 11T^{2} \) |
| 13 | \( 1 + 0.116T + 13T^{2} \) |
| 17 | \( 1 - 7.82T + 17T^{2} \) |
| 19 | \( 1 + 4.20T + 19T^{2} \) |
| 23 | \( 1 - 0.805T + 23T^{2} \) |
| 29 | \( 1 - 5.14T + 29T^{2} \) |
| 31 | \( 1 + 1.85T + 31T^{2} \) |
| 37 | \( 1 - 8.89T + 37T^{2} \) |
| 41 | \( 1 + 2.08T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 + 1.01T + 53T^{2} \) |
| 59 | \( 1 + 6.22T + 59T^{2} \) |
| 61 | \( 1 + 5.30T + 61T^{2} \) |
| 67 | \( 1 - 0.319T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 1.76T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 2.77T + 83T^{2} \) |
| 89 | \( 1 + 9.93T + 89T^{2} \) |
| 97 | \( 1 + 7.69T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.120510787001827448739500197055, −7.48608854763645120329345559990, −6.36687977459523164438001141500, −5.68494065804494932639551006956, −5.10723618559833206555069988514, −4.49279250559054764019461326836, −3.29062884627597798521326310186, −2.74179589055268806840692305958, −1.57385604644328883086985954181, 0,
1.57385604644328883086985954181, 2.74179589055268806840692305958, 3.29062884627597798521326310186, 4.49279250559054764019461326836, 5.10723618559833206555069988514, 5.68494065804494932639551006956, 6.36687977459523164438001141500, 7.48608854763645120329345559990, 8.120510787001827448739500197055
