| L(s) = 1 | + 1.06·2-s + 7.81·3-s − 6.86·4-s + 6.14·5-s + 8.32·6-s − 5.24·7-s − 15.8·8-s + 34.1·9-s + 6.53·10-s − 14.5·11-s − 53.6·12-s − 49.9·13-s − 5.58·14-s + 48.0·15-s + 38.0·16-s + 75.6·17-s + 36.3·18-s + 128.·19-s − 42.1·20-s − 41.0·21-s − 15.4·22-s − 131.·23-s − 123.·24-s − 87.2·25-s − 53.1·26-s + 55.8·27-s + 36.0·28-s + ⋯ |
| L(s) = 1 | + 0.376·2-s + 1.50·3-s − 0.858·4-s + 0.549·5-s + 0.566·6-s − 0.283·7-s − 0.699·8-s + 1.26·9-s + 0.206·10-s − 0.397·11-s − 1.29·12-s − 1.06·13-s − 0.106·14-s + 0.826·15-s + 0.595·16-s + 1.07·17-s + 0.475·18-s + 1.54·19-s − 0.471·20-s − 0.426·21-s − 0.149·22-s − 1.19·23-s − 1.05·24-s − 0.698·25-s − 0.400·26-s + 0.398·27-s + 0.243·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 1.06T + 8T^{2} \) |
| 3 | \( 1 - 7.81T + 27T^{2} \) |
| 5 | \( 1 - 6.14T + 125T^{2} \) |
| 7 | \( 1 + 5.24T + 343T^{2} \) |
| 11 | \( 1 + 14.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 75.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 128.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 36.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 264.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 0.0646T + 6.89e4T^{2} \) |
| 47 | \( 1 - 346.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 468.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 586.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 592.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 472.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 548.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 328.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 801.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.34e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 281.T + 9.12e5T^{2} \) |
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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
−8.596980698645103843333439585859, −7.70972188275777345597830504334, −7.30462894667235504912241041119, −5.71914806250836620979169046363, −5.32711166745733490600671305952, −4.09125381310200553280050321947, −3.39386615940578585045966712293, −2.66019338081243540569924929759, −1.59342264801720836345663641566, 0,
1.59342264801720836345663641566, 2.66019338081243540569924929759, 3.39386615940578585045966712293, 4.09125381310200553280050321947, 5.32711166745733490600671305952, 5.71914806250836620979169046363, 7.30462894667235504912241041119, 7.70972188275777345597830504334, 8.596980698645103843333439585859
