| L(s) = 1 | − 1.44·2-s + 2.93·3-s + 0.0803·4-s + 3.90·5-s − 4.23·6-s − 1.48·7-s + 2.76·8-s + 5.62·9-s − 5.62·10-s − 4.53·11-s + 0.236·12-s − 2.23·13-s + 2.14·14-s + 11.4·15-s − 4.15·16-s + 1.30·17-s − 8.11·18-s − 1.58·19-s + 0.313·20-s − 4.37·21-s + 6.53·22-s − 7.42·23-s + 8.13·24-s + 10.2·25-s + 3.22·26-s + 7.72·27-s − 0.119·28-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 1.69·3-s + 0.0401·4-s + 1.74·5-s − 1.72·6-s − 0.563·7-s + 0.978·8-s + 1.87·9-s − 1.77·10-s − 1.36·11-s + 0.0681·12-s − 0.620·13-s + 0.574·14-s + 2.95·15-s − 1.03·16-s + 0.316·17-s − 1.91·18-s − 0.364·19-s + 0.0701·20-s − 0.954·21-s + 1.39·22-s − 1.54·23-s + 1.66·24-s + 2.04·25-s + 0.632·26-s + 1.48·27-s − 0.0226·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.344677667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.344677667\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 227 | \( 1 - T \) |
| good | 2 | \( 1 + 1.44T + 2T^{2} \) |
| 3 | \( 1 - 2.93T + 3T^{2} \) |
| 5 | \( 1 - 3.90T + 5T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 11 | \( 1 + 4.53T + 11T^{2} \) |
| 13 | \( 1 + 2.23T + 13T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 + 1.58T + 19T^{2} \) |
| 23 | \( 1 + 7.42T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 - 5.84T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 2.46T + 41T^{2} \) |
| 43 | \( 1 + 4.58T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 + 2.47T + 53T^{2} \) |
| 59 | \( 1 + 6.01T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 5.60T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 6.91T + 79T^{2} \) |
| 83 | \( 1 - 7.01T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−12.72443317259840046065002796900, −10.43363694504084263312317806497, −9.784244502351739281238238310929, −9.581173155632292195309662994305, −8.365389772791061972000585728041, −7.79898261820255068575189271410, −6.37093240417619253879569313507, −4.77891714102536240696058825284, −2.84752772241843606094122912147, −1.93549336538828092950008409826,
1.93549336538828092950008409826, 2.84752772241843606094122912147, 4.77891714102536240696058825284, 6.37093240417619253879569313507, 7.79898261820255068575189271410, 8.365389772791061972000585728041, 9.581173155632292195309662994305, 9.784244502351739281238238310929, 10.43363694504084263312317806497, 12.72443317259840046065002796900
