| L(s) = 1 | + 4.31·2-s + 10.5·4-s − 7·7-s + 11.1·8-s + 12.8·11-s − 57.3·13-s − 30.1·14-s − 36.4·16-s + 17.1·17-s + 107.·19-s + 55.3·22-s − 67.1·23-s − 247.·26-s − 74.1·28-s + 52.7·29-s + 117.·31-s − 246.·32-s + 73.9·34-s − 150.·37-s + 464.·38-s − 396.·41-s − 431.·43-s + 135.·44-s − 289.·46-s − 389.·47-s + 49·49-s − 607.·52-s + ⋯ |
| L(s) = 1 | + 1.52·2-s + 1.32·4-s − 0.377·7-s + 0.494·8-s + 0.351·11-s − 1.22·13-s − 0.576·14-s − 0.570·16-s + 0.244·17-s + 1.30·19-s + 0.536·22-s − 0.608·23-s − 1.86·26-s − 0.500·28-s + 0.337·29-s + 0.679·31-s − 1.36·32-s + 0.372·34-s − 0.669·37-s + 1.98·38-s − 1.51·41-s − 1.53·43-s + 0.465·44-s − 0.928·46-s − 1.20·47-s + 0.142·49-s − 1.62·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 4.31T + 8T^{2} \) |
| 11 | \( 1 - 12.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 57.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 17.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 67.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 52.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 396.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 431.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 389.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 16.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 537.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 287.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 826.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 614.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 234.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 235.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 516.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.59e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.11e3T + 9.12e5T^{2} \) |
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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
−8.623732912730254198561519434228, −7.53539411623437738910352408360, −6.80371106709179097401715494696, −6.06721911854867696419476824412, −5.12810891188248061043234564322, −4.63866394655857219553264746174, −3.47171858164342712278257399814, −2.94666678576964040498848937050, −1.71827333648963244078954038449, 0,
1.71827333648963244078954038449, 2.94666678576964040498848937050, 3.47171858164342712278257399814, 4.63866394655857219553264746174, 5.12810891188248061043234564322, 6.06721911854867696419476824412, 6.80371106709179097401715494696, 7.53539411623437738910352408360, 8.623732912730254198561519434228
