| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 7-s + 3·8-s + 9-s + 12-s − 5·13-s − 14-s − 16-s + 8·17-s − 18-s − 19-s − 21-s + 6·23-s − 3·24-s + 5·26-s − 27-s − 28-s − 6·29-s + 3·31-s − 5·32-s − 8·34-s − 36-s + 9·37-s + 38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.288·12-s − 1.38·13-s − 0.267·14-s − 1/4·16-s + 1.94·17-s − 0.235·18-s − 0.229·19-s − 0.218·21-s + 1.25·23-s − 0.612·24-s + 0.980·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.538·31-s − 0.883·32-s − 1.37·34-s − 1/6·36-s + 1.47·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.57342529658568, −13.98902851073431, −13.49777988870593, −12.91919189016739, −12.37332110824003, −12.01548894998222, −11.45161759678218, −10.78282835009520, −10.35361687952685, −9.924026154811513, −9.412673418751803, −9.032395430315327, −8.210855794757134, −7.830047463232051, −7.283129537654144, −6.985885368355019, −5.947260625528474, −5.469963775125119, −4.963173160337026, −4.536367915554218, −3.797949507972739, −3.070031845209582, −2.255342684113992, −1.373704241597411, −0.8610080078077387, 0,
0.8610080078077387, 1.373704241597411, 2.255342684113992, 3.070031845209582, 3.797949507972739, 4.536367915554218, 4.963173160337026, 5.469963775125119, 5.947260625528474, 6.985885368355019, 7.283129537654144, 7.830047463232051, 8.210855794757134, 9.032395430315327, 9.412673418751803, 9.924026154811513, 10.35361687952685, 10.78282835009520, 11.45161759678218, 12.01548894998222, 12.37332110824003, 12.91919189016739, 13.49777988870593, 13.98902851073431, 14.57342529658568
