L(s) = 1 | + 1.73·2-s + 3-s + 0.999·4-s + 1.73·6-s + 7-s − 1.73·8-s + 9-s − 4·11-s + 0.999·12-s − 5.73·13-s + 1.73·14-s − 5·16-s − 3.19·17-s + 1.73·18-s + 1.73·19-s + 21-s − 6.92·22-s + 4.46·23-s − 1.73·24-s − 9.92·26-s + 27-s + 0.999·28-s + 0.464·29-s − 3.46·31-s − 5.19·32-s − 4·33-s − 5.53·34-s + ⋯ |
L(s) = 1 | + 1.22·2-s + 0.577·3-s + 0.499·4-s + 0.707·6-s + 0.377·7-s − 0.612·8-s + 0.333·9-s − 1.20·11-s + 0.288·12-s − 1.58·13-s + 0.462·14-s − 1.25·16-s − 0.775·17-s + 0.408·18-s + 0.397·19-s + 0.218·21-s − 1.47·22-s + 0.930·23-s − 0.353·24-s − 1.94·26-s + 0.192·27-s + 0.188·28-s + 0.0861·29-s − 0.622·31-s − 0.918·32-s − 0.696·33-s − 0.949·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 5.73T + 13T^{2} \) |
| 17 | \( 1 + 3.19T + 17T^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 - 4.46T + 23T^{2} \) |
| 29 | \( 1 - 0.464T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 6.46T + 41T^{2} \) |
| 43 | \( 1 - 4.53T + 43T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 15.1T + 59T^{2} \) |
| 61 | \( 1 + 5.92T + 61T^{2} \) |
| 67 | \( 1 - 8.92T + 67T^{2} \) |
| 71 | \( 1 - 3.19T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 5.46T + 79T^{2} \) |
| 83 | \( 1 - 2.92T + 83T^{2} \) |
| 89 | \( 1 + 6.53T + 89T^{2} \) |
| 97 | \( 1 - 5.46T + 97T^{2} \) |
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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
−8.123872592142635482826933870250, −7.29745520745495085110965811944, −6.78585614540459017253345756543, −5.54943705204085188520559461394, −5.01627813261020809880797425646, −4.54182584650991107472891690243, −3.44135174968241146884130943885, −2.75136717041583464148880957275, −2.00646790922642152845075487646, 0,
2.00646790922642152845075487646, 2.75136717041583464148880957275, 3.44135174968241146884130943885, 4.54182584650991107472891690243, 5.01627813261020809880797425646, 5.54943705204085188520559461394, 6.78585614540459017253345756543, 7.29745520745495085110965811944, 8.123872592142635482826933870250