| L(s) = 1 | − 2.10·2-s + 2.26·3-s + 2.44·4-s − 3.60·5-s − 4.77·6-s + 7-s − 0.948·8-s + 2.11·9-s + 7.59·10-s + 0.886·11-s + 5.53·12-s − 2.10·14-s − 8.14·15-s − 2.89·16-s − 4.96·17-s − 4.46·18-s + 2.37·19-s − 8.82·20-s + 2.26·21-s − 1.86·22-s − 3.85·23-s − 2.14·24-s + 7.97·25-s − 2.00·27-s + 2.44·28-s + 1.28·29-s + 17.1·30-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.30·3-s + 1.22·4-s − 1.61·5-s − 1.94·6-s + 0.377·7-s − 0.335·8-s + 0.705·9-s + 2.40·10-s + 0.267·11-s + 1.59·12-s − 0.563·14-s − 2.10·15-s − 0.724·16-s − 1.20·17-s − 1.05·18-s + 0.545·19-s − 1.97·20-s + 0.493·21-s − 0.398·22-s − 0.804·23-s − 0.437·24-s + 1.59·25-s − 0.385·27-s + 0.462·28-s + 0.238·29-s + 3.13·30-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 | \( 1 + 2.10T + 2T^{2} \) |
| 3 | \( 1 - 2.26T + 3T^{2} \) |
| 5 | \( 1 + 3.60T + 5T^{2} \) |
| 11 | \( 1 - 0.886T + 11T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 + 3.85T + 23T^{2} \) |
| 29 | \( 1 - 1.28T + 29T^{2} \) |
| 31 | \( 1 - 8.46T + 31T^{2} \) |
| 37 | \( 1 + 9.63T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 + 3.64T + 43T^{2} \) |
| 47 | \( 1 - 2.98T + 47T^{2} \) |
| 53 | \( 1 - 4.92T + 53T^{2} \) |
| 59 | \( 1 + 7.32T + 59T^{2} \) |
| 61 | \( 1 + 1.53T + 61T^{2} \) |
| 67 | \( 1 + 8.42T + 67T^{2} \) |
| 71 | \( 1 + 6.44T + 71T^{2} \) |
| 73 | \( 1 + 7.14T + 73T^{2} \) |
| 79 | \( 1 - 0.757T + 79T^{2} \) |
| 83 | \( 1 + 4.76T + 83T^{2} \) |
| 89 | \( 1 - 3.61T + 89T^{2} \) |
| 97 | \( 1 + 0.463T + 97T^{2} \) |
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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
−8.878774877893289786586120666951, −8.633280657092037276389406426530, −7.994995938730088763431235389001, −7.40918186695774763695536922720, −6.65200496899691311837260772000, −4.71352894134357188277774433537, −3.86711441138319156372196123897, −2.84414500039436555285677234307, −1.62732053820000223381133027002, 0,
1.62732053820000223381133027002, 2.84414500039436555285677234307, 3.86711441138319156372196123897, 4.71352894134357188277774433537, 6.65200496899691311837260772000, 7.40918186695774763695536922720, 7.994995938730088763431235389001, 8.633280657092037276389406426530, 8.878774877893289786586120666951
