| L(s) = 1 | − 1.87·2-s − 2.53·3-s + 1.53·4-s − 0.120·5-s + 4.75·6-s + 1.53·7-s + 0.879·8-s + 3.41·9-s + 0.226·10-s − 2.69·11-s − 3.87·12-s + 4.57·13-s − 2.87·14-s + 0.305·15-s − 4.71·16-s − 6.41·18-s − 1.87·19-s − 0.184·20-s − 3.87·21-s + 5.06·22-s − 7.41·23-s − 2.22·24-s − 4.98·25-s − 8.59·26-s − 1.04·27-s + 2.34·28-s − 3.41·29-s + ⋯ |
| L(s) = 1 | − 1.32·2-s − 1.46·3-s + 0.766·4-s − 0.0539·5-s + 1.94·6-s + 0.579·7-s + 0.310·8-s + 1.13·9-s + 0.0716·10-s − 0.812·11-s − 1.11·12-s + 1.26·13-s − 0.769·14-s + 0.0788·15-s − 1.17·16-s − 1.51·18-s − 0.431·19-s − 0.0413·20-s − 0.846·21-s + 1.07·22-s − 1.54·23-s − 0.454·24-s − 0.997·25-s − 1.68·26-s − 0.200·27-s + 0.443·28-s − 0.633·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 + 1.87T + 2T^{2} \) |
| 3 | \( 1 + 2.53T + 3T^{2} \) |
| 5 | \( 1 + 0.120T + 5T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 + 2.69T + 11T^{2} \) |
| 13 | \( 1 - 4.57T + 13T^{2} \) |
| 19 | \( 1 + 1.87T + 19T^{2} \) |
| 23 | \( 1 + 7.41T + 23T^{2} \) |
| 29 | \( 1 + 3.41T + 29T^{2} \) |
| 31 | \( 1 + 3.83T + 31T^{2} \) |
| 37 | \( 1 - 5.24T + 37T^{2} \) |
| 41 | \( 1 + 6.24T + 41T^{2} \) |
| 43 | \( 1 - 5.49T + 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 - 0.822T + 53T^{2} \) |
| 59 | \( 1 + 3.14T + 59T^{2} \) |
| 61 | \( 1 + 1.22T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 0.170T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 4.14T + 79T^{2} \) |
| 83 | \( 1 + 2.43T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−11.08093019875477368241328088586, −10.54188644395724906876774334374, −9.622388178437547886730676546442, −8.358947973327930343732013226224, −7.70151217019981360479092873290, −6.40487929196159375792237509287, −5.50607060691888397723522168374, −4.24319968810377895626514480258, −1.67019056885653638230218909251, 0,
1.67019056885653638230218909251, 4.24319968810377895626514480258, 5.50607060691888397723522168374, 6.40487929196159375792237509287, 7.70151217019981360479092873290, 8.358947973327930343732013226224, 9.622388178437547886730676546442, 10.54188644395724906876774334374, 11.08093019875477368241328088586
