| L(s) = 1 | + 2-s + 4-s − 2.42·5-s − 4.62·7-s + 8-s − 2.42·10-s − 5.75·11-s − 5.37·13-s − 4.62·14-s + 16-s − 2.76·17-s + 3.14·19-s − 2.42·20-s − 5.75·22-s + 1.27·23-s + 0.862·25-s − 5.37·26-s − 4.62·28-s + 0.713·29-s − 3.11·31-s + 32-s − 2.76·34-s + 11.1·35-s − 4.81·37-s + 3.14·38-s − 2.42·40-s − 9.85·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.08·5-s − 1.74·7-s + 0.353·8-s − 0.765·10-s − 1.73·11-s − 1.49·13-s − 1.23·14-s + 0.250·16-s − 0.671·17-s + 0.720·19-s − 0.541·20-s − 1.22·22-s + 0.266·23-s + 0.172·25-s − 1.05·26-s − 0.873·28-s + 0.132·29-s − 0.560·31-s + 0.176·32-s − 0.475·34-s + 1.89·35-s − 0.791·37-s + 0.509·38-s − 0.382·40-s − 1.53·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2280525262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2280525262\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 2.42T + 5T^{2} \) |
| 7 | \( 1 + 4.62T + 7T^{2} \) |
| 11 | \( 1 + 5.75T + 11T^{2} \) |
| 13 | \( 1 + 5.37T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 - 3.14T + 19T^{2} \) |
| 23 | \( 1 - 1.27T + 23T^{2} \) |
| 29 | \( 1 - 0.713T + 29T^{2} \) |
| 31 | \( 1 + 3.11T + 31T^{2} \) |
| 37 | \( 1 + 4.81T + 37T^{2} \) |
| 41 | \( 1 + 9.85T + 41T^{2} \) |
| 43 | \( 1 - 0.0559T + 43T^{2} \) |
| 47 | \( 1 + 8.45T + 47T^{2} \) |
| 53 | \( 1 - 2.03T + 53T^{2} \) |
| 59 | \( 1 + 4.28T + 59T^{2} \) |
| 61 | \( 1 + 5.69T + 61T^{2} \) |
| 67 | \( 1 + 4.54T + 67T^{2} \) |
| 71 | \( 1 - 12.0T + 71T^{2} \) |
| 73 | \( 1 + 4.39T + 73T^{2} \) |
| 79 | \( 1 - 8.53T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + 5.81T + 89T^{2} \) |
| 97 | \( 1 - 6.97T + 97T^{2} \) |
| 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
−7.60401591997351822967753640888, −7.10252993249926993524262876722, −6.58852431439635178204840354610, −5.55754087410590827126660071547, −5.04545867528724220922478337091, −4.29614178644026297530741016494, −3.26937428085148930698784989683, −3.06326164058003253856478787157, −2.15617133789119122424074711427, −0.19133838295445139470549316619,
0.19133838295445139470549316619, 2.15617133789119122424074711427, 3.06326164058003253856478787157, 3.26937428085148930698784989683, 4.29614178644026297530741016494, 5.04545867528724220922478337091, 5.55754087410590827126660071547, 6.58852431439635178204840354610, 7.10252993249926993524262876722, 7.60401591997351822967753640888
