| L(s) = 1 | − 1.81·2-s + 3-s + 1.27·4-s − 2.99·5-s − 1.81·6-s − 4.54·7-s + 1.31·8-s + 9-s + 5.42·10-s − 11-s + 1.27·12-s + 8.22·14-s − 2.99·15-s − 4.92·16-s + 5.59·17-s − 1.81·18-s − 0.429·19-s − 3.82·20-s − 4.54·21-s + 1.81·22-s − 7.72·23-s + 1.31·24-s + 3.98·25-s + 27-s − 5.79·28-s + 4.56·29-s + 5.42·30-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.577·3-s + 0.638·4-s − 1.34·5-s − 0.738·6-s − 1.71·7-s + 0.463·8-s + 0.333·9-s + 1.71·10-s − 0.301·11-s + 0.368·12-s + 2.19·14-s − 0.773·15-s − 1.23·16-s + 1.35·17-s − 0.426·18-s − 0.0985·19-s − 0.855·20-s − 0.991·21-s + 0.385·22-s − 1.60·23-s + 0.267·24-s + 0.796·25-s + 0.192·27-s − 1.09·28-s + 0.848·29-s + 0.990·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 5 | \( 1 + 2.99T + 5T^{2} \) |
| 7 | \( 1 + 4.54T + 7T^{2} \) |
| 17 | \( 1 - 5.59T + 17T^{2} \) |
| 19 | \( 1 + 0.429T + 19T^{2} \) |
| 23 | \( 1 + 7.72T + 23T^{2} \) |
| 29 | \( 1 - 4.56T + 29T^{2} \) |
| 31 | \( 1 + 0.104T + 31T^{2} \) |
| 37 | \( 1 - 8.29T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 - 2.28T + 43T^{2} \) |
| 47 | \( 1 - 9.86T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 8.83T + 59T^{2} \) |
| 61 | \( 1 + 5.46T + 61T^{2} \) |
| 67 | \( 1 + 6.76T + 67T^{2} \) |
| 71 | \( 1 - 2.50T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 7.11T + 79T^{2} \) |
| 83 | \( 1 - 0.999T + 83T^{2} \) |
| 89 | \( 1 + 7.25T + 89T^{2} \) |
| 97 | \( 1 - 9.47T + 97T^{2} \) |
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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
−7.87888837585165602850074248408, −7.47696347944633549492914805213, −6.68690459638588387827231065273, −5.93360461793180148889613035324, −4.63937453760578752492182730183, −3.79140230788478790663200451691, −3.28787118830132591060662404958, −2.31966338192145284610418124081, −0.885944425181942093893893299411, 0,
0.885944425181942093893893299411, 2.31966338192145284610418124081, 3.28787118830132591060662404958, 3.79140230788478790663200451691, 4.63937453760578752492182730183, 5.93360461793180148889613035324, 6.68690459638588387827231065273, 7.47696347944633549492914805213, 7.87888837585165602850074248408
