| L(s) = 1 | + 2.50·2-s + 3·3-s − 1.72·4-s + 8.04·5-s + 7.51·6-s + 1.07·7-s − 24.3·8-s + 9·9-s + 20.1·10-s + 11·11-s − 5.17·12-s − 45.7·13-s + 2.69·14-s + 24.1·15-s − 47.2·16-s + 93.6·17-s + 22.5·18-s + 7.12·19-s − 13.8·20-s + 3.23·21-s + 27.5·22-s − 154.·23-s − 73.0·24-s − 60.3·25-s − 114.·26-s + 27·27-s − 1.85·28-s + ⋯ |
| L(s) = 1 | + 0.885·2-s + 0.577·3-s − 0.215·4-s + 0.719·5-s + 0.511·6-s + 0.0581·7-s − 1.07·8-s + 0.333·9-s + 0.636·10-s + 0.301·11-s − 0.124·12-s − 0.976·13-s + 0.0515·14-s + 0.415·15-s − 0.738·16-s + 1.33·17-s + 0.295·18-s + 0.0860·19-s − 0.154·20-s + 0.0335·21-s + 0.267·22-s − 1.40·23-s − 0.621·24-s − 0.482·25-s − 0.864·26-s + 0.192·27-s − 0.0125·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 2.50T + 8T^{2} \) |
| 5 | \( 1 - 8.04T + 125T^{2} \) |
| 7 | \( 1 - 1.07T + 343T^{2} \) |
| 13 | \( 1 + 45.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 93.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7.12T + 6.85e3T^{2} \) |
| 23 | \( 1 + 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 6.26T + 2.43e4T^{2} \) |
| 31 | \( 1 + 229.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 49.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 325.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 95.0T + 7.95e4T^{2} \) |
| 47 | \( 1 - 247.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 111.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 294.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 501.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 567.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 912.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 455.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 540.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 439.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.33e3T + 9.12e5T^{2} \) |
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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
−8.391656330441344287288923214193, −7.65077648683621691656354752363, −6.69540557949951573479217125620, −5.69022566843129478700120769804, −5.29382526188024284514021617787, −4.19690957904046487638651118698, −3.52046872372530689115878696968, −2.55937500474749437364477070376, −1.59339664949740973448876637033, 0,
1.59339664949740973448876637033, 2.55937500474749437364477070376, 3.52046872372530689115878696968, 4.19690957904046487638651118698, 5.29382526188024284514021617787, 5.69022566843129478700120769804, 6.69540557949951573479217125620, 7.65077648683621691656354752363, 8.391656330441344287288923214193
