| L(s) = 1 | + 4.15·2-s − 14.7·4-s + 37.5·5-s + 76.4·7-s − 194.·8-s + 155.·10-s − 169.·13-s + 317.·14-s − 333.·16-s − 0.875·17-s + 817.·19-s − 553.·20-s − 749.·23-s − 1.71e3·25-s − 704.·26-s − 1.12e3·28-s + 6.04e3·29-s − 1.47e3·31-s + 4.82e3·32-s − 3.63·34-s + 2.86e3·35-s − 1.58e4·37-s + 3.39e3·38-s − 7.28e3·40-s − 7.62e3·41-s + 1.82e4·43-s − 3.11e3·46-s + ⋯ |
| L(s) = 1 | + 0.733·2-s − 0.461·4-s + 0.671·5-s + 0.589·7-s − 1.07·8-s + 0.492·10-s − 0.278·13-s + 0.433·14-s − 0.325·16-s − 0.000735·17-s + 0.519·19-s − 0.309·20-s − 0.295·23-s − 0.549·25-s − 0.204·26-s − 0.272·28-s + 1.33·29-s − 0.275·31-s + 0.833·32-s − 0.000539·34-s + 0.395·35-s − 1.90·37-s + 0.381·38-s − 0.719·40-s − 0.708·41-s + 1.50·43-s − 0.216·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 4.15T + 32T^{2} \) |
| 5 | \( 1 - 37.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 76.4T + 1.68e4T^{2} \) |
| 13 | \( 1 + 169.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 0.875T + 1.41e6T^{2} \) |
| 19 | \( 1 - 817.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 749.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.04e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.47e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.58e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.62e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.82e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.28e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.17e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.16e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.75e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.46e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
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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
−8.767151463784952009832596887816, −7.982033124386967914680627867900, −6.87650364266963120293445321422, −5.90280863771653536160831281477, −5.24022665836716432204096674087, −4.50668723564895017112291623640, −3.51217677068655899075410597566, −2.46556985567324476534855689533, −1.32865949041461137251133219424, 0,
1.32865949041461137251133219424, 2.46556985567324476534855689533, 3.51217677068655899075410597566, 4.50668723564895017112291623640, 5.24022665836716432204096674087, 5.90280863771653536160831281477, 6.87650364266963120293445321422, 7.982033124386967914680627867900, 8.767151463784952009832596887816
