| L(s) = 1 | − 2.55·2-s + 1.15·3-s + 4.53·4-s + 4.10·5-s − 2.94·6-s − 4.64·7-s − 6.46·8-s − 1.67·9-s − 10.5·10-s − 11-s + 5.21·12-s + 0.666·13-s + 11.8·14-s + 4.73·15-s + 7.46·16-s + 4.42·17-s + 4.27·18-s + 6.64·19-s + 18.6·20-s − 5.34·21-s + 2.55·22-s − 1.40·23-s − 7.44·24-s + 11.8·25-s − 1.70·26-s − 5.38·27-s − 21.0·28-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 0.664·3-s + 2.26·4-s + 1.83·5-s − 1.20·6-s − 1.75·7-s − 2.28·8-s − 0.558·9-s − 3.32·10-s − 0.301·11-s + 1.50·12-s + 0.184·13-s + 3.17·14-s + 1.22·15-s + 1.86·16-s + 1.07·17-s + 1.00·18-s + 1.52·19-s + 4.16·20-s − 1.16·21-s + 0.544·22-s − 0.293·23-s − 1.51·24-s + 2.37·25-s − 0.334·26-s − 1.03·27-s − 3.97·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9449150405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9449150405\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 3 | \( 1 - 1.15T + 3T^{2} \) |
| 5 | \( 1 - 4.10T + 5T^{2} \) |
| 7 | \( 1 + 4.64T + 7T^{2} \) |
| 13 | \( 1 - 0.666T + 13T^{2} \) |
| 17 | \( 1 - 4.42T + 17T^{2} \) |
| 19 | \( 1 - 6.64T + 19T^{2} \) |
| 23 | \( 1 + 1.40T + 23T^{2} \) |
| 29 | \( 1 - 5.66T + 29T^{2} \) |
| 31 | \( 1 - 5.31T + 31T^{2} \) |
| 37 | \( 1 - 4.06T + 37T^{2} \) |
| 41 | \( 1 - 6.02T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 1.84T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 2.44T + 59T^{2} \) |
| 67 | \( 1 - 7.44T + 67T^{2} \) |
| 71 | \( 1 + 3.85T + 71T^{2} \) |
| 73 | \( 1 + 8.62T + 73T^{2} \) |
| 79 | \( 1 + 4.96T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 0.290T + 89T^{2} \) |
| 97 | \( 1 - 8.07T + 97T^{2} \) |
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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
−9.946826410421998728360696610929, −9.592901781464092413693609686635, −9.178014040777368530730036985886, −8.170739521846455224314891712740, −7.18660940799916422970779357900, −6.17136429720569423063122398796, −5.75028811253602641197816323050, −2.97797735153434968112679033141, −2.64343494774103471017350241236, −1.08412789459222287153254725270,
1.08412789459222287153254725270, 2.64343494774103471017350241236, 2.97797735153434968112679033141, 5.75028811253602641197816323050, 6.17136429720569423063122398796, 7.18660940799916422970779357900, 8.170739521846455224314891712740, 9.178014040777368530730036985886, 9.592901781464092413693609686635, 9.946826410421998728360696610929
