| L(s) = 1 | + 1.35·2-s − 3-s − 0.161·4-s − 1.20·5-s − 1.35·6-s + 1.50·7-s − 2.93·8-s + 9-s − 1.63·10-s − 0.378·11-s + 0.161·12-s + 0.475·13-s + 2.03·14-s + 1.20·15-s − 3.65·16-s + 17-s + 1.35·18-s − 0.243·19-s + 0.194·20-s − 1.50·21-s − 0.513·22-s + 3.98·23-s + 2.93·24-s − 3.54·25-s + 0.644·26-s − 27-s − 0.242·28-s + ⋯ |
| L(s) = 1 | + 0.958·2-s − 0.577·3-s − 0.0807·4-s − 0.538·5-s − 0.553·6-s + 0.568·7-s − 1.03·8-s + 0.333·9-s − 0.516·10-s − 0.114·11-s + 0.0466·12-s + 0.131·13-s + 0.544·14-s + 0.310·15-s − 0.912·16-s + 0.242·17-s + 0.319·18-s − 0.0558·19-s + 0.0435·20-s − 0.328·21-s − 0.109·22-s + 0.831·23-s + 0.598·24-s − 0.709·25-s + 0.126·26-s − 0.192·27-s − 0.0459·28-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 | \( 1 - 1.35T + 2T^{2} \) |
| 5 | \( 1 + 1.20T + 5T^{2} \) |
| 7 | \( 1 - 1.50T + 7T^{2} \) |
| 11 | \( 1 + 0.378T + 11T^{2} \) |
| 13 | \( 1 - 0.475T + 13T^{2} \) |
| 19 | \( 1 + 0.243T + 19T^{2} \) |
| 23 | \( 1 - 3.98T + 23T^{2} \) |
| 29 | \( 1 - 0.911T + 29T^{2} \) |
| 31 | \( 1 - 5.95T + 31T^{2} \) |
| 37 | \( 1 + 6.86T + 37T^{2} \) |
| 41 | \( 1 + 3.78T + 41T^{2} \) |
| 43 | \( 1 + 0.424T + 43T^{2} \) |
| 47 | \( 1 + 0.0605T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + 1.99T + 59T^{2} \) |
| 61 | \( 1 + 3.38T + 61T^{2} \) |
| 67 | \( 1 - 8.87T + 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 - 6.41T + 73T^{2} \) |
| 79 | \( 1 - 9.64T + 79T^{2} \) |
| 83 | \( 1 - 1.79T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
| 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
−7.34665113715852908935745263915, −6.62834710139589128733708425071, −5.93555602207750534728234839044, −5.17082231530000813058363045940, −4.80500660348490013676217937947, −4.00372187641834529518649050265, −3.41626534877694218142073390098, −2.44124823308608880376817364732, −1.18622015938678563401719839669, 0,
1.18622015938678563401719839669, 2.44124823308608880376817364732, 3.41626534877694218142073390098, 4.00372187641834529518649050265, 4.80500660348490013676217937947, 5.17082231530000813058363045940, 5.93555602207750534728234839044, 6.62834710139589128733708425071, 7.34665113715852908935745263915
