| L(s) = 1 | − 2-s − 1.73·3-s + 4-s − 5-s + 1.73·6-s − 7-s − 8-s + 10-s − 1.73·12-s + 2.26·13-s + 14-s + 1.73·15-s + 16-s − 2.73·17-s + 3.73·19-s − 20-s + 1.73·21-s − 5·23-s + 1.73·24-s + 25-s − 2.26·26-s + 5.19·27-s − 28-s − 8·29-s − 1.73·30-s + 0.732·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.00·3-s + 0.5·4-s − 0.447·5-s + 0.707·6-s − 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.500·12-s + 0.629·13-s + 0.267·14-s + 0.447·15-s + 0.250·16-s − 0.662·17-s + 0.856·19-s − 0.223·20-s + 0.377·21-s − 1.04·23-s + 0.353·24-s + 0.200·25-s − 0.444·26-s + 1.00·27-s − 0.188·28-s − 1.48·29-s − 0.316·30-s + 0.131·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 13 | \( 1 - 2.26T + 13T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 19 | \( 1 - 3.73T + 19T^{2} \) |
| 23 | \( 1 + 5T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 - 0.732T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 4.19T + 41T^{2} \) |
| 43 | \( 1 - 8.73T + 43T^{2} \) |
| 47 | \( 1 + 1.26T + 47T^{2} \) |
| 53 | \( 1 - 4.19T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 7.66T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 7T + 79T^{2} \) |
| 83 | \( 1 + 5T + 83T^{2} \) |
| 89 | \( 1 - 3.66T + 89T^{2} \) |
| 97 | \( 1 + 2.92T + 97T^{2} \) |
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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
−7.54066265604383766733640420031, −6.70053608739809852477112962485, −6.09313845145714771319559003235, −5.64434103993468672148402415541, −4.71321165744082534783722713024, −3.85556968358981354744066863022, −3.06243181597717704350034581675, −2.01281206688825571003090974479, −0.881100865411984934444298050466, 0,
0.881100865411984934444298050466, 2.01281206688825571003090974479, 3.06243181597717704350034581675, 3.85556968358981354744066863022, 4.71321165744082534783722713024, 5.64434103993468672148402415541, 6.09313845145714771319559003235, 6.70053608739809852477112962485, 7.54066265604383766733640420031
