| L(s) = 1 | − 2.76·2-s + 0.315·3-s + 5.65·4-s + 0.739·5-s − 0.874·6-s + 4.71·7-s − 10.1·8-s − 2.90·9-s − 2.04·10-s − 4.47·11-s + 1.78·12-s − 13.0·14-s + 0.233·15-s + 16.6·16-s − 1.20·17-s + 8.02·18-s + 4.53·19-s + 4.18·20-s + 1.49·21-s + 12.3·22-s + 23-s − 3.19·24-s − 4.45·25-s − 1.86·27-s + 26.6·28-s + 1.63·29-s − 0.646·30-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 0.182·3-s + 2.82·4-s + 0.330·5-s − 0.356·6-s + 1.78·7-s − 3.57·8-s − 0.966·9-s − 0.646·10-s − 1.34·11-s + 0.515·12-s − 3.48·14-s + 0.0602·15-s + 4.17·16-s − 0.291·17-s + 1.89·18-s + 1.03·19-s + 0.935·20-s + 0.325·21-s + 2.64·22-s + 0.208·23-s − 0.652·24-s − 0.890·25-s − 0.358·27-s + 5.04·28-s + 0.303·29-s − 0.117·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3887 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3887 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8492388785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8492388785\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.76T + 2T^{2} \) |
| 3 | \( 1 - 0.315T + 3T^{2} \) |
| 5 | \( 1 - 0.739T + 5T^{2} \) |
| 7 | \( 1 - 4.71T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 17 | \( 1 + 1.20T + 17T^{2} \) |
| 19 | \( 1 - 4.53T + 19T^{2} \) |
| 29 | \( 1 - 1.63T + 29T^{2} \) |
| 31 | \( 1 + 2.02T + 31T^{2} \) |
| 37 | \( 1 + 1.30T + 37T^{2} \) |
| 41 | \( 1 + 5.73T + 41T^{2} \) |
| 43 | \( 1 + 3.32T + 43T^{2} \) |
| 47 | \( 1 + 8.46T + 47T^{2} \) |
| 53 | \( 1 - 8.80T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 3.37T + 61T^{2} \) |
| 67 | \( 1 - 7.84T + 67T^{2} \) |
| 71 | \( 1 - 1.11T + 71T^{2} \) |
| 73 | \( 1 - 2.83T + 73T^{2} \) |
| 79 | \( 1 - 6.85T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 6.44T + 89T^{2} \) |
| 97 | \( 1 + 3.68T + 97T^{2} \) |
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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
−8.377674509861507111869007153642, −8.003715340972482239012905311532, −7.49951488014182487199213287069, −6.60398115910358305128885182915, −5.47048366042593365992120747067, −5.19788608452894472782088916467, −3.38266624669078946043546699660, −2.34408229934363144990166300422, −1.90615490833834134786059150288, −0.69166465490565267393505328678,
0.69166465490565267393505328678, 1.90615490833834134786059150288, 2.34408229934363144990166300422, 3.38266624669078946043546699660, 5.19788608452894472782088916467, 5.47048366042593365992120747067, 6.60398115910358305128885182915, 7.49951488014182487199213287069, 8.003715340972482239012905311532, 8.377674509861507111869007153642
