| L(s) = 1 | − 2.30·2-s + 1.87·3-s + 3.33·4-s − 0.652·5-s − 4.33·6-s − 3.07·8-s + 0.518·9-s + 1.50·10-s + 0.394·11-s + 6.24·12-s − 6.61·13-s − 1.22·15-s + 0.433·16-s + 17-s − 1.19·18-s − 0.739·19-s − 2.17·20-s − 0.910·22-s + 0.223·23-s − 5.76·24-s − 4.57·25-s + 15.2·26-s − 4.65·27-s − 5.35·29-s + 2.82·30-s − 4.25·31-s + 5.14·32-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.08·3-s + 1.66·4-s − 0.291·5-s − 1.76·6-s − 1.08·8-s + 0.172·9-s + 0.476·10-s + 0.118·11-s + 1.80·12-s − 1.83·13-s − 0.315·15-s + 0.108·16-s + 0.242·17-s − 0.282·18-s − 0.169·19-s − 0.485·20-s − 0.194·22-s + 0.0466·23-s − 1.17·24-s − 0.914·25-s + 2.99·26-s − 0.895·27-s − 0.994·29-s + 0.515·30-s − 0.764·31-s + 0.909·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 - 1.87T + 3T^{2} \) |
| 5 | \( 1 + 0.652T + 5T^{2} \) |
| 11 | \( 1 - 0.394T + 11T^{2} \) |
| 13 | \( 1 + 6.61T + 13T^{2} \) |
| 19 | \( 1 + 0.739T + 19T^{2} \) |
| 23 | \( 1 - 0.223T + 23T^{2} \) |
| 29 | \( 1 + 5.35T + 29T^{2} \) |
| 31 | \( 1 + 4.25T + 31T^{2} \) |
| 37 | \( 1 + 9.01T + 37T^{2} \) |
| 41 | \( 1 - 9.79T + 41T^{2} \) |
| 43 | \( 1 - 7.13T + 43T^{2} \) |
| 47 | \( 1 + 4.57T + 47T^{2} \) |
| 53 | \( 1 - 2.03T + 53T^{2} \) |
| 59 | \( 1 - 8.80T + 59T^{2} \) |
| 61 | \( 1 + 9.79T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 0.0440T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 9.80T + 89T^{2} \) |
| 97 | \( 1 + 0.652T + 97T^{2} \) |
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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
−9.449309801019550863941159836522, −9.128957952879717744478564911342, −8.106178660777343109241856812134, −7.60231780709917793027203451868, −6.98469190780293741395237314614, −5.50373590842502153352310206238, −4.03194993980328496417223999485, −2.72541307192515306679627019070, −1.90185217549445537457691545154, 0,
1.90185217549445537457691545154, 2.72541307192515306679627019070, 4.03194993980328496417223999485, 5.50373590842502153352310206238, 6.98469190780293741395237314614, 7.60231780709917793027203451868, 8.106178660777343109241856812134, 9.128957952879717744478564911342, 9.449309801019550863941159836522
