| L(s) = 1 | + 0.231·2-s − 3.32·3-s − 1.94·4-s − 2.23·5-s − 0.768·6-s − 7-s − 0.913·8-s + 8.03·9-s − 0.516·10-s + 3.32·11-s + 6.46·12-s − 0.231·14-s + 7.41·15-s + 3.68·16-s − 1.37·17-s + 1.85·18-s + 3.23·19-s + 4.34·20-s + 3.32·21-s + 0.768·22-s + 0.838·23-s + 3.03·24-s − 0.0210·25-s − 16.7·27-s + 1.94·28-s − 0.607·29-s + 1.71·30-s + ⋯ |
| L(s) = 1 | + 0.163·2-s − 1.91·3-s − 0.973·4-s − 0.997·5-s − 0.313·6-s − 0.377·7-s − 0.322·8-s + 2.67·9-s − 0.163·10-s + 1.00·11-s + 1.86·12-s − 0.0618·14-s + 1.91·15-s + 0.920·16-s − 0.333·17-s + 0.438·18-s + 0.742·19-s + 0.971·20-s + 0.724·21-s + 0.163·22-s + 0.174·23-s + 0.619·24-s − 0.00420·25-s − 3.22·27-s + 0.367·28-s − 0.112·29-s + 0.313·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.231T + 2T^{2} \) |
| 3 | \( 1 + 3.32T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 11 | \( 1 - 3.32T + 11T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 - 0.838T + 23T^{2} \) |
| 29 | \( 1 + 0.607T + 29T^{2} \) |
| 31 | \( 1 - 1.71T + 31T^{2} \) |
| 37 | \( 1 - 1.55T + 37T^{2} \) |
| 41 | \( 1 + 9.17T + 41T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 + 1.62T + 47T^{2} \) |
| 53 | \( 1 - 8.39T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 5.21T + 71T^{2} \) |
| 73 | \( 1 + 3.96T + 73T^{2} \) |
| 79 | \( 1 - 6.45T + 79T^{2} \) |
| 83 | \( 1 + 4.64T + 83T^{2} \) |
| 89 | \( 1 - 9.12T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−9.588439911299603120206967342882, −8.594244504606833345595898884760, −7.45188317751329422908022627397, −6.72265073755264883690083529313, −5.86166744812408625635242639380, −5.06030148911497938702384519597, −4.25509348511142810547265904703, −3.63747894231620138915707622226, −1.10147507383469668706023969506, 0,
1.10147507383469668706023969506, 3.63747894231620138915707622226, 4.25509348511142810547265904703, 5.06030148911497938702384519597, 5.86166744812408625635242639380, 6.72265073755264883690083529313, 7.45188317751329422908022627397, 8.594244504606833345595898884760, 9.588439911299603120206967342882
