| L(s) = 1 | + 23.7·2-s + 50.4·4-s + 2.75e3·5-s − 8.23e3·7-s − 1.09e4·8-s + 6.54e4·10-s − 7.77e4·11-s + 2.22e4·13-s − 1.95e5·14-s − 2.85e5·16-s − 2.79e5·17-s − 2.82e5·19-s + 1.39e5·20-s − 1.84e6·22-s + 5.73e5·23-s + 5.65e6·25-s + 5.28e5·26-s − 4.15e5·28-s − 2.50e6·29-s − 4.00e6·31-s − 1.16e6·32-s − 6.62e6·34-s − 2.27e7·35-s + 7.25e5·37-s − 6.70e6·38-s − 3.01e7·40-s − 1.88e7·41-s + ⋯ |
| L(s) = 1 | + 1.04·2-s + 0.0985·4-s + 1.97·5-s − 1.29·7-s − 0.944·8-s + 2.06·10-s − 1.60·11-s + 0.216·13-s − 1.35·14-s − 1.08·16-s − 0.811·17-s − 0.497·19-s + 0.194·20-s − 1.67·22-s + 0.427·23-s + 2.89·25-s + 0.226·26-s − 0.127·28-s − 0.658·29-s − 0.778·31-s − 0.196·32-s − 0.850·34-s − 2.55·35-s + 0.0636·37-s − 0.521·38-s − 1.86·40-s − 1.04·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 23.7T + 512T^{2} \) |
| 5 | \( 1 - 2.75e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 8.23e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.77e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.22e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.79e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.82e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 5.73e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.50e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.00e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 7.25e5T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.88e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 8.38e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.96e6T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.37e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.07e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.40e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 9.76e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.01e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 5.36e6T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.85e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.36e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.37e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.11e8T + 7.60e17T^{2} \) |
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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
−12.74006560821965189276905826136, −10.71575193336890451533966525229, −9.762604983078064351758922405641, −8.901577577327959308257187808427, −6.66917353055784781458920030061, −5.83961215569022687987732246723, −4.99117634337825505343376026512, −3.15991720095668085369693804831, −2.20445491728070375229398836672, 0,
2.20445491728070375229398836672, 3.15991720095668085369693804831, 4.99117634337825505343376026512, 5.83961215569022687987732246723, 6.66917353055784781458920030061, 8.901577577327959308257187808427, 9.762604983078064351758922405641, 10.71575193336890451533966525229, 12.74006560821965189276905826136
