| L(s) = 1 | + 1.61·2-s + 0.618·4-s + 0.381·5-s − 3·7-s − 2.23·8-s + 0.618·10-s − 6.23·13-s − 4.85·14-s − 4.85·16-s − 0.618·17-s + 0.854·19-s + 0.236·20-s + 5.47·23-s − 4.85·25-s − 10.0·26-s − 1.85·28-s − 4.47·29-s − 3.85·31-s − 3.38·32-s − 1.00·34-s − 1.14·35-s − 4.23·37-s + 1.38·38-s − 0.854·40-s + 5.94·41-s − 1.76·43-s + 8.85·46-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.309·4-s + 0.170·5-s − 1.13·7-s − 0.790·8-s + 0.195·10-s − 1.72·13-s − 1.29·14-s − 1.21·16-s − 0.149·17-s + 0.195·19-s + 0.0527·20-s + 1.14·23-s − 0.970·25-s − 1.97·26-s − 0.350·28-s − 0.830·29-s − 0.692·31-s − 0.597·32-s − 0.171·34-s − 0.193·35-s − 0.696·37-s + 0.224·38-s − 0.135·40-s + 0.928·41-s − 0.268·43-s + 1.30·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 - 0.381T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 + 0.618T + 17T^{2} \) |
| 19 | \( 1 - 0.854T + 19T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 - 5.94T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 - 0.618T + 47T^{2} \) |
| 53 | \( 1 - 7.38T + 53T^{2} \) |
| 59 | \( 1 - 5.32T + 59T^{2} \) |
| 61 | \( 1 + 1.14T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 + 1.23T + 73T^{2} \) |
| 79 | \( 1 + 0.527T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + 9.47T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−9.524096781083806482405267963682, −8.870081131059040310806460206012, −7.42851311613229906114255433378, −6.81937082769566968630102068145, −5.78804420872436102615481721321, −5.16956713587085711223667600466, −4.16233951598199570839061576865, −3.23433371526089713122833376846, −2.36794852029282235412265183818, 0,
2.36794852029282235412265183818, 3.23433371526089713122833376846, 4.16233951598199570839061576865, 5.16956713587085711223667600466, 5.78804420872436102615481721321, 6.81937082769566968630102068145, 7.42851311613229906114255433378, 8.870081131059040310806460206012, 9.524096781083806482405267963682
