L(s) = 1 | − 2-s + 0.980·3-s + 4-s − 0.685·5-s − 0.980·6-s − 1.46·7-s − 8-s − 2.03·9-s + 0.685·10-s + 0.0958·11-s + 0.980·12-s − 5.45·13-s + 1.46·14-s − 0.672·15-s + 16-s − 6.05·17-s + 2.03·18-s + 5.10·19-s − 0.685·20-s − 1.43·21-s − 0.0958·22-s − 4.13·23-s − 0.980·24-s − 4.52·25-s + 5.45·26-s − 4.94·27-s − 1.46·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.566·3-s + 0.5·4-s − 0.306·5-s − 0.400·6-s − 0.552·7-s − 0.353·8-s − 0.679·9-s + 0.216·10-s + 0.0289·11-s + 0.283·12-s − 1.51·13-s + 0.390·14-s − 0.173·15-s + 0.250·16-s − 1.46·17-s + 0.480·18-s + 1.17·19-s − 0.153·20-s − 0.312·21-s − 0.0204·22-s − 0.862·23-s − 0.200·24-s − 0.905·25-s + 1.06·26-s − 0.951·27-s − 0.276·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 269 | \( 1 + T \) |
good | 3 | \( 1 - 0.980T + 3T^{2} \) |
| 5 | \( 1 + 0.685T + 5T^{2} \) |
| 7 | \( 1 + 1.46T + 7T^{2} \) |
| 11 | \( 1 - 0.0958T + 11T^{2} \) |
| 13 | \( 1 + 5.45T + 13T^{2} \) |
| 17 | \( 1 + 6.05T + 17T^{2} \) |
| 19 | \( 1 - 5.10T + 19T^{2} \) |
| 23 | \( 1 + 4.13T + 23T^{2} \) |
| 29 | \( 1 - 5.03T + 29T^{2} \) |
| 31 | \( 1 - 1.86T + 31T^{2} \) |
| 37 | \( 1 - 9.97T + 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 + 2.42T + 43T^{2} \) |
| 47 | \( 1 + 3.13T + 47T^{2} \) |
| 53 | \( 1 + 6.72T + 53T^{2} \) |
| 59 | \( 1 - 5.30T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 6.38T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 5.54T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 2.28T + 97T^{2} \) |
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
−9.952704404298776188549141123199, −9.625057528633788924438493198650, −8.582060928394737458020005394005, −7.85716393945938051458077367939, −6.98385039008245990505615479113, −5.95472762095435739740519685514, −4.58852126074377547156857133849, −3.16758831231193328187824031299, −2.25944495664960247894814222884, 0,
2.25944495664960247894814222884, 3.16758831231193328187824031299, 4.58852126074377547156857133849, 5.95472762095435739740519685514, 6.98385039008245990505615479113, 7.85716393945938051458077367939, 8.582060928394737458020005394005, 9.625057528633788924438493198650, 9.952704404298776188549141123199