L(s) = 1 | + 2-s − 2.61·3-s + 4-s − 1.38·5-s − 2.61·6-s − 3·7-s + 8-s + 3.85·9-s − 1.38·10-s − 2.38·11-s − 2.61·12-s − 3·13-s − 3·14-s + 3.61·15-s + 16-s − 7.47·17-s + 3.85·18-s − 3·19-s − 1.38·20-s + 7.85·21-s − 2.38·22-s + 1.47·23-s − 2.61·24-s − 3.09·25-s − 3·26-s − 2.23·27-s − 3·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.51·3-s + 0.5·4-s − 0.618·5-s − 1.06·6-s − 1.13·7-s + 0.353·8-s + 1.28·9-s − 0.437·10-s − 0.718·11-s − 0.755·12-s − 0.832·13-s − 0.801·14-s + 0.934·15-s + 0.250·16-s − 1.81·17-s + 0.908·18-s − 0.688·19-s − 0.309·20-s + 1.71·21-s − 0.507·22-s + 0.306·23-s − 0.534·24-s − 0.618·25-s − 0.588·26-s − 0.430·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6038 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6038 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3019 | \( 1+O(T) \) |
good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 2.38T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + 7.47T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 - 1.47T + 23T^{2} \) |
| 29 | \( 1 + 9.23T + 29T^{2} \) |
| 31 | \( 1 + 6.09T + 31T^{2} \) |
| 37 | \( 1 + 1.85T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 9.23T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + 9.76T + 59T^{2} \) |
| 61 | \( 1 + 2.38T + 61T^{2} \) |
| 67 | \( 1 - 1.85T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 0.708T + 73T^{2} \) |
| 79 | \( 1 + 1.85T + 79T^{2} \) |
| 83 | \( 1 + 1.94T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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
−7.20193873697552341772487463710, −6.45082350381719454975173900019, −5.93825411806962806963479187343, −5.25737649908674734961429262711, −4.49115166715489506444535199749, −3.94922063529863278727306577464, −2.89493040810100742624055535855, −1.94434150521789333176918400896, 0, 0,
1.94434150521789333176918400896, 2.89493040810100742624055535855, 3.94922063529863278727306577464, 4.49115166715489506444535199749, 5.25737649908674734961429262711, 5.93825411806962806963479187343, 6.45082350381719454975173900019, 7.20193873697552341772487463710