| L(s) = 1 | − 0.0826·2-s − 3-s − 1.99·4-s + 1.50·5-s + 0.0826·6-s − 7-s + 0.330·8-s + 9-s − 0.124·10-s − 4.66·11-s + 1.99·12-s + 0.536·13-s + 0.0826·14-s − 1.50·15-s + 3.95·16-s − 6.79·17-s − 0.0826·18-s + 5.44·19-s − 3.00·20-s + 21-s + 0.385·22-s + 2.01·23-s − 0.330·24-s − 2.73·25-s − 0.0443·26-s − 27-s + 1.99·28-s + ⋯ |
| L(s) = 1 | − 0.0584·2-s − 0.577·3-s − 0.996·4-s + 0.673·5-s + 0.0337·6-s − 0.377·7-s + 0.116·8-s + 0.333·9-s − 0.0393·10-s − 1.40·11-s + 0.575·12-s + 0.148·13-s + 0.0220·14-s − 0.388·15-s + 0.989·16-s − 1.64·17-s − 0.0194·18-s + 1.24·19-s − 0.671·20-s + 0.218·21-s + 0.0821·22-s + 0.420·23-s − 0.0673·24-s − 0.546·25-s − 0.00868·26-s − 0.192·27-s + 0.376·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0826T + 2T^{2} \) |
| 5 | \( 1 - 1.50T + 5T^{2} \) |
| 11 | \( 1 + 4.66T + 11T^{2} \) |
| 13 | \( 1 - 0.536T + 13T^{2} \) |
| 17 | \( 1 + 6.79T + 17T^{2} \) |
| 19 | \( 1 - 5.44T + 19T^{2} \) |
| 23 | \( 1 - 2.01T + 23T^{2} \) |
| 29 | \( 1 - 5.34T + 29T^{2} \) |
| 31 | \( 1 + 1.38T + 31T^{2} \) |
| 37 | \( 1 - 1.81T + 37T^{2} \) |
| 41 | \( 1 - 6.61T + 41T^{2} \) |
| 43 | \( 1 + 1.14T + 43T^{2} \) |
| 47 | \( 1 - 5.60T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 59 | \( 1 + 3.13T + 59T^{2} \) |
| 61 | \( 1 + 0.293T + 61T^{2} \) |
| 67 | \( 1 - 4.93T + 67T^{2} \) |
| 71 | \( 1 + 2.65T + 71T^{2} \) |
| 73 | \( 1 - 0.237T + 73T^{2} \) |
| 79 | \( 1 + 9.45T + 79T^{2} \) |
| 83 | \( 1 - 6.94T + 83T^{2} \) |
| 89 | \( 1 + 6.07T + 89T^{2} \) |
| 97 | \( 1 + 3.72T + 97T^{2} \) |
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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
−7.49387396537013874400600361016, −6.75007808685416863201002050279, −5.86139278836289458692752661741, −5.46035883156563336586603124516, −4.75635392405949312455570960918, −4.12650346434875631206533132149, −3.04864446343663378293589738620, −2.25968233179326327732854008313, −0.993344355775283308302971109147, 0,
0.993344355775283308302971109147, 2.25968233179326327732854008313, 3.04864446343663378293589738620, 4.12650346434875631206533132149, 4.75635392405949312455570960918, 5.46035883156563336586603124516, 5.86139278836289458692752661741, 6.75007808685416863201002050279, 7.49387396537013874400600361016
