| L(s) = 1 | − 2-s + 0.796·3-s + 4-s + 1.51·5-s − 0.796·6-s − 4.16·7-s − 8-s − 2.36·9-s − 1.51·10-s + 11-s + 0.796·12-s + 2.21·13-s + 4.16·14-s + 1.21·15-s + 16-s − 5.06·17-s + 2.36·18-s + 2.21·19-s + 1.51·20-s − 3.31·21-s − 22-s + 4.95·23-s − 0.796·24-s − 2.69·25-s − 2.21·26-s − 4.27·27-s − 4.16·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.460·3-s + 0.5·4-s + 0.679·5-s − 0.325·6-s − 1.57·7-s − 0.353·8-s − 0.788·9-s − 0.480·10-s + 0.301·11-s + 0.230·12-s + 0.613·13-s + 1.11·14-s + 0.312·15-s + 0.250·16-s − 1.22·17-s + 0.557·18-s + 0.507·19-s + 0.339·20-s − 0.723·21-s − 0.213·22-s + 1.03·23-s − 0.162·24-s − 0.538·25-s − 0.433·26-s − 0.822·27-s − 0.786·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 0.796T + 3T^{2} \) |
| 5 | \( 1 - 1.51T + 5T^{2} \) |
| 7 | \( 1 + 4.16T + 7T^{2} \) |
| 13 | \( 1 - 2.21T + 13T^{2} \) |
| 17 | \( 1 + 5.06T + 17T^{2} \) |
| 19 | \( 1 - 2.21T + 19T^{2} \) |
| 23 | \( 1 - 4.95T + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 + 7.95T + 31T^{2} \) |
| 37 | \( 1 - 5.42T + 37T^{2} \) |
| 41 | \( 1 + 7.10T + 41T^{2} \) |
| 43 | \( 1 + 3.96T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 59 | \( 1 + 3.99T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 5.89T + 71T^{2} \) |
| 73 | \( 1 + 8.75T + 73T^{2} \) |
| 79 | \( 1 - 8.62T + 79T^{2} \) |
| 83 | \( 1 - 8.68T + 83T^{2} \) |
| 89 | \( 1 - 6.91T + 89T^{2} \) |
| 97 | \( 1 + 7.15T + 97T^{2} \) |
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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
−9.265716159376731508732242104839, −8.934411126058347491727700154367, −7.85898133883153401900565531980, −6.76797923555119263440439178699, −6.27309097136080314810158131473, −5.37828915465684691291084285110, −3.66815198558964571314598225829, −2.96547998146480441558875849659, −1.84443033377186410770565270359, 0,
1.84443033377186410770565270359, 2.96547998146480441558875849659, 3.66815198558964571314598225829, 5.37828915465684691291084285110, 6.27309097136080314810158131473, 6.76797923555119263440439178699, 7.85898133883153401900565531980, 8.934411126058347491727700154367, 9.265716159376731508732242104839
