L(s) = 1 | + 0.618·2-s − 1.61·4-s − 2.85·7-s − 2.23·8-s + 1.61·11-s + 3.47·13-s − 1.76·14-s + 1.85·16-s − 7.47·17-s + 5.85·19-s + 1.00·22-s − 4.85·23-s + 2.14·26-s + 4.61·28-s + 7.23·29-s + 2·31-s + 5.61·32-s − 4.61·34-s + 3·37-s + 3.61·38-s + 2.47·41-s + 8.47·43-s − 2.61·44-s − 3.00·46-s − 9.38·47-s + 1.14·49-s − 5.61·52-s + ⋯ |
L(s) = 1 | + 0.437·2-s − 0.809·4-s − 1.07·7-s − 0.790·8-s + 0.487·11-s + 0.962·13-s − 0.471·14-s + 0.463·16-s − 1.81·17-s + 1.34·19-s + 0.213·22-s − 1.01·23-s + 0.420·26-s + 0.872·28-s + 1.34·29-s + 0.359·31-s + 0.993·32-s − 0.791·34-s + 0.493·37-s + 0.586·38-s + 0.386·41-s + 1.29·43-s − 0.394·44-s − 0.442·46-s − 1.36·47-s + 0.163·49-s − 0.779·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 + 7.47T + 17T^{2} \) |
| 19 | \( 1 - 5.85T + 19T^{2} \) |
| 23 | \( 1 + 4.85T + 23T^{2} \) |
| 29 | \( 1 - 7.23T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 + 9.38T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 + 3.94T + 59T^{2} \) |
| 61 | \( 1 + 2.14T + 61T^{2} \) |
| 67 | \( 1 + 2.52T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + 6.85T + 73T^{2} \) |
| 79 | \( 1 - 6.70T + 79T^{2} \) |
| 83 | \( 1 - 4.94T + 83T^{2} \) |
| 89 | \( 1 + 1.38T + 89T^{2} \) |
| 97 | \( 1 + 8.38T + 97T^{2} \) |
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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
−7.88853698352647366217674120195, −6.79827079079622531146088883879, −6.26546482847243600976778341612, −5.75022788251178262451060567220, −4.65565157931635942122052662935, −4.15187376658563593192546674036, −3.36921134534008706796559270764, −2.65776609986496083070525828405, −1.18586945317934901529615323259, 0,
1.18586945317934901529615323259, 2.65776609986496083070525828405, 3.36921134534008706796559270764, 4.15187376658563593192546674036, 4.65565157931635942122052662935, 5.75022788251178262451060567220, 6.26546482847243600976778341612, 6.79827079079622531146088883879, 7.88853698352647366217674120195