| L(s) = 1 | − 0.312·2-s + 1.10·3-s − 1.90·4-s − 1.93·5-s − 0.343·6-s + 7-s + 1.21·8-s − 1.78·9-s + 0.603·10-s + 3.30·11-s − 2.09·12-s − 0.312·14-s − 2.12·15-s + 3.42·16-s − 2.68·17-s + 0.558·18-s + 3.97·19-s + 3.68·20-s + 1.10·21-s − 1.03·22-s − 0.249·23-s + 1.34·24-s − 1.25·25-s − 5.27·27-s − 1.90·28-s − 6.85·29-s + 0.664·30-s + ⋯ |
| L(s) = 1 | − 0.220·2-s + 0.635·3-s − 0.951·4-s − 0.865·5-s − 0.140·6-s + 0.377·7-s + 0.430·8-s − 0.596·9-s + 0.190·10-s + 0.996·11-s − 0.604·12-s − 0.0834·14-s − 0.549·15-s + 0.856·16-s − 0.650·17-s + 0.131·18-s + 0.911·19-s + 0.823·20-s + 0.240·21-s − 0.219·22-s − 0.0520·23-s + 0.273·24-s − 0.251·25-s − 1.01·27-s − 0.359·28-s − 1.27·29-s + 0.121·30-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 | \( 1 + 0.312T + 2T^{2} \) |
| 3 | \( 1 - 1.10T + 3T^{2} \) |
| 5 | \( 1 + 1.93T + 5T^{2} \) |
| 11 | \( 1 - 3.30T + 11T^{2} \) |
| 17 | \( 1 + 2.68T + 17T^{2} \) |
| 19 | \( 1 - 3.97T + 19T^{2} \) |
| 23 | \( 1 + 0.249T + 23T^{2} \) |
| 29 | \( 1 + 6.85T + 29T^{2} \) |
| 31 | \( 1 + 8.16T + 31T^{2} \) |
| 37 | \( 1 - 1.51T + 37T^{2} \) |
| 41 | \( 1 + 1.36T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 - 2.89T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 2.82T + 67T^{2} \) |
| 71 | \( 1 - 9.17T + 71T^{2} \) |
| 73 | \( 1 + 8.47T + 73T^{2} \) |
| 79 | \( 1 + 8.68T + 79T^{2} \) |
| 83 | \( 1 + 8.89T + 83T^{2} \) |
| 89 | \( 1 - 7.50T + 89T^{2} \) |
| 97 | \( 1 - 8.70T + 97T^{2} \) |
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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
−9.171686345263249552423806245843, −8.633316809866705087302854521329, −7.87563231177587876994929374617, −7.24396399716768192207386691728, −5.88291114200086949852440884551, −4.90285014430311740308920988579, −3.90931170615413242494446834323, −3.36198429017099241016754447383, −1.69442318639432463619272445310, 0,
1.69442318639432463619272445310, 3.36198429017099241016754447383, 3.90931170615413242494446834323, 4.90285014430311740308920988579, 5.88291114200086949852440884551, 7.24396399716768192207386691728, 7.87563231177587876994929374617, 8.633316809866705087302854521329, 9.171686345263249552423806245843
