L(s) = 1 | + 1.72·3-s − 1.54·5-s + 7-s − 0.0353·9-s − 4.07·11-s − 0.958·13-s − 2.65·15-s + 1.53·17-s + 19-s + 1.72·21-s + 8.92·23-s − 2.62·25-s − 5.22·27-s + 2.51·29-s + 0.464·31-s − 7.02·33-s − 1.54·35-s + 6.79·37-s − 1.64·39-s + 1.97·41-s − 6.20·43-s + 0.0544·45-s + 2.22·47-s + 49-s + 2.64·51-s − 10.3·53-s + 6.29·55-s + ⋯ |
L(s) = 1 | + 0.994·3-s − 0.689·5-s + 0.377·7-s − 0.0117·9-s − 1.22·11-s − 0.265·13-s − 0.685·15-s + 0.372·17-s + 0.229·19-s + 0.375·21-s + 1.86·23-s − 0.524·25-s − 1.00·27-s + 0.467·29-s + 0.0833·31-s − 1.22·33-s − 0.260·35-s + 1.11·37-s − 0.264·39-s + 0.309·41-s − 0.945·43-s + 0.00811·45-s + 0.324·47-s + 0.142·49-s + 0.370·51-s − 1.42·53-s + 0.848·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 1.72T + 3T^{2} \) |
| 5 | \( 1 + 1.54T + 5T^{2} \) |
| 11 | \( 1 + 4.07T + 11T^{2} \) |
| 13 | \( 1 + 0.958T + 13T^{2} \) |
| 17 | \( 1 - 1.53T + 17T^{2} \) |
| 23 | \( 1 - 8.92T + 23T^{2} \) |
| 29 | \( 1 - 2.51T + 29T^{2} \) |
| 31 | \( 1 - 0.464T + 31T^{2} \) |
| 37 | \( 1 - 6.79T + 37T^{2} \) |
| 41 | \( 1 - 1.97T + 41T^{2} \) |
| 43 | \( 1 + 6.20T + 43T^{2} \) |
| 47 | \( 1 - 2.22T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 3.97T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + 9.47T + 73T^{2} \) |
| 79 | \( 1 - 2.03T + 79T^{2} \) |
| 83 | \( 1 - 1.26T + 83T^{2} \) |
| 89 | \( 1 - 2.48T + 89T^{2} \) |
| 97 | \( 1 - 2.57T + 97T^{2} \) |
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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
−7.79481785219182158808775584694, −7.04794099470978254497192555976, −6.03454781887597721217680295204, −5.17988250537088290064817969464, −4.65684395091472690375357463850, −3.70492965600462767200895612279, −2.94592039475595099166437861894, −2.54216203082875004985271026795, −1.32868249510858826107593507675, 0,
1.32868249510858826107593507675, 2.54216203082875004985271026795, 2.94592039475595099166437861894, 3.70492965600462767200895612279, 4.65684395091472690375357463850, 5.17988250537088290064817969464, 6.03454781887597721217680295204, 7.04794099470978254497192555976, 7.79481785219182158808775584694