| L(s) = 1 | − 0.454·3-s + 0.622·5-s + 4.14·7-s − 2.79·9-s − 1.04·11-s + 4.67·13-s − 0.282·15-s + 1.91·17-s − 6.57·19-s − 1.88·21-s + 2.26·23-s − 4.61·25-s + 2.63·27-s − 9.03·29-s + 1.46·31-s + 0.476·33-s + 2.58·35-s + 6.96·37-s − 2.12·39-s − 0.613·41-s + 3.34·43-s − 1.73·45-s + 2.36·47-s + 10.1·49-s − 0.872·51-s + 0.528·53-s − 0.652·55-s + ⋯ |
| L(s) = 1 | − 0.262·3-s + 0.278·5-s + 1.56·7-s − 0.931·9-s − 0.315·11-s + 1.29·13-s − 0.0730·15-s + 0.465·17-s − 1.50·19-s − 0.411·21-s + 0.472·23-s − 0.922·25-s + 0.506·27-s − 1.67·29-s + 0.263·31-s + 0.0828·33-s + 0.436·35-s + 1.14·37-s − 0.339·39-s − 0.0957·41-s + 0.509·43-s − 0.259·45-s + 0.345·47-s + 1.45·49-s − 0.122·51-s + 0.0725·53-s − 0.0879·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.223243156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.223243156\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 0.454T + 3T^{2} \) |
| 5 | \( 1 - 0.622T + 5T^{2} \) |
| 7 | \( 1 - 4.14T + 7T^{2} \) |
| 11 | \( 1 + 1.04T + 11T^{2} \) |
| 13 | \( 1 - 4.67T + 13T^{2} \) |
| 17 | \( 1 - 1.91T + 17T^{2} \) |
| 19 | \( 1 + 6.57T + 19T^{2} \) |
| 23 | \( 1 - 2.26T + 23T^{2} \) |
| 29 | \( 1 + 9.03T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 - 6.96T + 37T^{2} \) |
| 41 | \( 1 + 0.613T + 41T^{2} \) |
| 43 | \( 1 - 3.34T + 43T^{2} \) |
| 47 | \( 1 - 2.36T + 47T^{2} \) |
| 53 | \( 1 - 0.528T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 4.74T + 61T^{2} \) |
| 67 | \( 1 - 16.2T + 67T^{2} \) |
| 71 | \( 1 + 8.46T + 71T^{2} \) |
| 73 | \( 1 - 6.36T + 73T^{2} \) |
| 79 | \( 1 - 2.85T + 79T^{2} \) |
| 83 | \( 1 - 1.00T + 83T^{2} \) |
| 89 | \( 1 + 13.9T + 89T^{2} \) |
| 97 | \( 1 - 5.06T + 97T^{2} \) |
| 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
−8.082748671740092752860334974123, −7.20593622067494314538771818317, −6.23422001577180206364113669572, −5.69989506138915493333863428989, −5.21463784304866582378922195953, −4.28537017975449078868328868325, −3.66357700269107229639136967485, −2.44676542993389112251760908735, −1.82255384994257141550725431528, −0.75987469614199259913304411153,
0.75987469614199259913304411153, 1.82255384994257141550725431528, 2.44676542993389112251760908735, 3.66357700269107229639136967485, 4.28537017975449078868328868325, 5.21463784304866582378922195953, 5.69989506138915493333863428989, 6.23422001577180206364113669572, 7.20593622067494314538771818317, 8.082748671740092752860334974123
