| L(s) = 1 | − 0.845·2-s + 0.0216·3-s − 1.28·4-s − 2.12·5-s − 0.0183·6-s + 2.77·8-s − 2.99·9-s + 1.79·10-s − 5.19·11-s − 0.0278·12-s − 3.20·13-s − 0.0461·15-s + 0.219·16-s − 6.26·17-s + 2.53·18-s + 1.56·19-s + 2.73·20-s + 4.39·22-s − 4.17·23-s + 0.0602·24-s − 0.473·25-s + 2.71·26-s − 0.130·27-s + 3.92·29-s + 0.0390·30-s + 3.32·31-s − 5.74·32-s + ⋯ |
| L(s) = 1 | − 0.598·2-s + 0.0125·3-s − 0.642·4-s − 0.951·5-s − 0.00748·6-s + 0.982·8-s − 0.999·9-s + 0.569·10-s − 1.56·11-s − 0.00803·12-s − 0.889·13-s − 0.0119·15-s + 0.0548·16-s − 1.52·17-s + 0.597·18-s + 0.358·19-s + 0.611·20-s + 0.936·22-s − 0.871·23-s + 0.0122·24-s − 0.0947·25-s + 0.532·26-s − 0.0250·27-s + 0.729·29-s + 0.00712·30-s + 0.597·31-s − 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1609183098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1609183098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 0.845T + 2T^{2} \) |
| 3 | \( 1 - 0.0216T + 3T^{2} \) |
| 5 | \( 1 + 2.12T + 5T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 + 3.20T + 13T^{2} \) |
| 17 | \( 1 + 6.26T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 + 4.17T + 23T^{2} \) |
| 29 | \( 1 - 3.92T + 29T^{2} \) |
| 31 | \( 1 - 3.32T + 31T^{2} \) |
| 37 | \( 1 - 0.150T + 37T^{2} \) |
| 43 | \( 1 + 4.19T + 43T^{2} \) |
| 47 | \( 1 + 0.154T + 47T^{2} \) |
| 53 | \( 1 + 6.90T + 53T^{2} \) |
| 59 | \( 1 - 0.905T + 59T^{2} \) |
| 61 | \( 1 + 7.22T + 61T^{2} \) |
| 67 | \( 1 + 1.72T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 - 0.331T + 79T^{2} \) |
| 83 | \( 1 + 6.16T + 83T^{2} \) |
| 89 | \( 1 + 1.82T + 89T^{2} \) |
| 97 | \( 1 - 7.73T + 97T^{2} \) |
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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
−9.017779947803697430101604337315, −8.227252449031042548685685744448, −7.943077925154293328160617571655, −7.13445669400268289623750945497, −5.92785608907109027790821703379, −4.91292289220339468142595370198, −4.44310003253718719956131312350, −3.20947361935546128249237142049, −2.24001380619520437526352310793, −0.27355404721401864843912463704,
0.27355404721401864843912463704, 2.24001380619520437526352310793, 3.20947361935546128249237142049, 4.44310003253718719956131312350, 4.91292289220339468142595370198, 5.92785608907109027790821703379, 7.13445669400268289623750945497, 7.943077925154293328160617571655, 8.227252449031042548685685744448, 9.017779947803697430101604337315
