| L(s) = 1 | − 2.08·2-s + 1.49·3-s + 2.34·4-s + 2.98·5-s − 3.11·6-s + 3.78·7-s − 0.718·8-s − 0.771·9-s − 6.21·10-s + 11-s + 3.50·12-s + 4.41·13-s − 7.89·14-s + 4.45·15-s − 3.19·16-s + 5.30·17-s + 1.60·18-s + 0.311·19-s + 6.99·20-s + 5.65·21-s − 2.08·22-s − 5.62·23-s − 1.07·24-s + 3.89·25-s − 9.20·26-s − 5.63·27-s + 8.87·28-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 0.861·3-s + 1.17·4-s + 1.33·5-s − 1.27·6-s + 1.43·7-s − 0.254·8-s − 0.257·9-s − 1.96·10-s + 0.301·11-s + 1.01·12-s + 1.22·13-s − 2.10·14-s + 1.14·15-s − 0.797·16-s + 1.28·17-s + 0.378·18-s + 0.0715·19-s + 1.56·20-s + 1.23·21-s − 0.444·22-s − 1.17·23-s − 0.218·24-s + 0.778·25-s − 1.80·26-s − 1.08·27-s + 1.67·28-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
| \(L(1)\) |
\(\approx\) |
\(1.426713793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.426713793\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.08T + 2T^{2} \) |
| 3 | \( 1 - 1.49T + 3T^{2} \) |
| 5 | \( 1 - 2.98T + 5T^{2} \) |
| 7 | \( 1 - 3.78T + 7T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 17 | \( 1 - 5.30T + 17T^{2} \) |
| 19 | \( 1 - 0.311T + 19T^{2} \) |
| 23 | \( 1 + 5.62T + 23T^{2} \) |
| 29 | \( 1 + 8.30T + 29T^{2} \) |
| 31 | \( 1 + 6.85T + 31T^{2} \) |
| 37 | \( 1 - 4.98T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 0.135T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 9.36T + 53T^{2} \) |
| 59 | \( 1 + 7.08T + 59T^{2} \) |
| 67 | \( 1 - 3.08T + 67T^{2} \) |
| 71 | \( 1 - 4.02T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−10.17913481738466794621262701441, −9.483400356186973534072419111293, −8.799216289436909926552262344255, −8.143632919599986007154819194428, −7.55296070431939772337207759591, −6.14964952829298315494090023257, −5.29023128196851630087973172895, −3.63123479032662860250173018008, −1.99516978746505766591956325254, −1.53442379267755864628283863673,
1.53442379267755864628283863673, 1.99516978746505766591956325254, 3.63123479032662860250173018008, 5.29023128196851630087973172895, 6.14964952829298315494090023257, 7.55296070431939772337207759591, 8.143632919599986007154819194428, 8.799216289436909926552262344255, 9.483400356186973534072419111293, 10.17913481738466794621262701441
