| L(s) = 1 | + 0.618·2-s − 3-s − 1.61·4-s + 1.61·5-s − 0.618·6-s − 7-s − 2.23·8-s + 9-s + 1.00·10-s − 2.23·11-s + 1.61·12-s − 4.61·13-s − 0.618·14-s − 1.61·15-s + 1.85·16-s − 6.70·17-s + 0.618·18-s + 5.47·19-s − 2.61·20-s + 21-s − 1.38·22-s − 23-s + 2.23·24-s − 2.38·25-s − 2.85·26-s − 27-s + 1.61·28-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.577·3-s − 0.809·4-s + 0.723·5-s − 0.252·6-s − 0.377·7-s − 0.790·8-s + 0.333·9-s + 0.316·10-s − 0.674·11-s + 0.467·12-s − 1.28·13-s − 0.165·14-s − 0.417·15-s + 0.463·16-s − 1.62·17-s + 0.145·18-s + 1.25·19-s − 0.585·20-s + 0.218·21-s − 0.294·22-s − 0.208·23-s + 0.456·24-s − 0.476·25-s − 0.559·26-s − 0.192·27-s + 0.305·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 + 6.70T + 17T^{2} \) |
| 19 | \( 1 - 5.47T + 19T^{2} \) |
| 29 | \( 1 + 3.76T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 - 7.47T + 41T^{2} \) |
| 43 | \( 1 - 0.618T + 43T^{2} \) |
| 47 | \( 1 + 2.76T + 47T^{2} \) |
| 53 | \( 1 + 1.90T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 1.85T + 61T^{2} \) |
| 67 | \( 1 - 6.09T + 67T^{2} \) |
| 71 | \( 1 - 6.61T + 71T^{2} \) |
| 73 | \( 1 - 0.708T + 73T^{2} \) |
| 79 | \( 1 + 0.527T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 9.38T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
| 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
−10.43456108302839949315555401966, −9.619949858350752214505865533901, −9.075972103146551047298056541198, −7.67789560682717400194453991327, −6.65306542256103135596342541307, −5.48259355845956839893593043248, −5.06715251370191373343897543879, −3.79473129818647837080905150654, −2.30587730490221570561396220636, 0,
2.30587730490221570561396220636, 3.79473129818647837080905150654, 5.06715251370191373343897543879, 5.48259355845956839893593043248, 6.65306542256103135596342541307, 7.67789560682717400194453991327, 9.075972103146551047298056541198, 9.619949858350752214505865533901, 10.43456108302839949315555401966
