| L(s) = 1 | − 0.347·2-s + 0.532·3-s − 1.87·4-s − 0.120·5-s − 0.184·6-s + 1.34·8-s − 2.71·9-s + 0.0418·10-s + 11-s − 12-s + 1.22·13-s − 0.0641·15-s + 3.29·16-s − 6.17·17-s + 0.943·18-s − 6.41·19-s + 0.226·20-s − 0.347·22-s − 2.02·23-s + 0.716·24-s − 4.98·25-s − 0.426·26-s − 3.04·27-s + 3.24·29-s + 0.0222·30-s − 4.87·31-s − 3.83·32-s + ⋯ |
| L(s) = 1 | − 0.245·2-s + 0.307·3-s − 0.939·4-s − 0.0539·5-s − 0.0754·6-s + 0.476·8-s − 0.905·9-s + 0.0132·10-s + 0.301·11-s − 0.288·12-s + 0.340·13-s − 0.0165·15-s + 0.822·16-s − 1.49·17-s + 0.222·18-s − 1.47·19-s + 0.0506·20-s − 0.0740·22-s − 0.421·23-s + 0.146·24-s − 0.997·25-s − 0.0835·26-s − 0.585·27-s + 0.603·29-s + 0.00406·30-s − 0.876·31-s − 0.678·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.347T + 2T^{2} \) |
| 3 | \( 1 - 0.532T + 3T^{2} \) |
| 5 | \( 1 + 0.120T + 5T^{2} \) |
| 13 | \( 1 - 1.22T + 13T^{2} \) |
| 17 | \( 1 + 6.17T + 17T^{2} \) |
| 19 | \( 1 + 6.41T + 19T^{2} \) |
| 23 | \( 1 + 2.02T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 + 4.87T + 31T^{2} \) |
| 37 | \( 1 - 2.36T + 37T^{2} \) |
| 41 | \( 1 + 8.29T + 41T^{2} \) |
| 43 | \( 1 - 2.22T + 43T^{2} \) |
| 47 | \( 1 + 9.23T + 47T^{2} \) |
| 53 | \( 1 - 9.68T + 53T^{2} \) |
| 59 | \( 1 - 9.74T + 59T^{2} \) |
| 61 | \( 1 - 1.43T + 61T^{2} \) |
| 67 | \( 1 + 3.06T + 67T^{2} \) |
| 71 | \( 1 + 8.49T + 71T^{2} \) |
| 73 | \( 1 + 3.53T + 73T^{2} \) |
| 79 | \( 1 - 9.09T + 79T^{2} \) |
| 83 | \( 1 - 7.02T + 83T^{2} \) |
| 89 | \( 1 - 6.87T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−10.30354098092292850522524301362, −9.271649993187301508655976240563, −8.617771835052148950722066138874, −8.105818149364681398750812646620, −6.71800240148044012355568790191, −5.72629622331668169229432660056, −4.51730874632868226647756268351, −3.67128842101921818844759233194, −2.11051481085416196848589283584, 0,
2.11051481085416196848589283584, 3.67128842101921818844759233194, 4.51730874632868226647756268351, 5.72629622331668169229432660056, 6.71800240148044012355568790191, 8.105818149364681398750812646620, 8.617771835052148950722066138874, 9.271649993187301508655976240563, 10.30354098092292850522524301362
