| L(s) = 1 | + 24.8·2-s + 107.·4-s + 625·5-s − 4.01e3·7-s − 1.00e4·8-s + 1.55e4·10-s − 8.48e4·11-s + 1.19e5·13-s − 9.97e4·14-s − 3.05e5·16-s − 1.16e5·17-s − 2.34e5·19-s + 6.70e4·20-s − 2.11e6·22-s − 2.34e6·23-s + 3.90e5·25-s + 2.97e6·26-s − 4.29e5·28-s + 4.64e5·29-s − 5.11e6·31-s − 2.44e6·32-s − 2.90e6·34-s − 2.50e6·35-s + 8.69e6·37-s − 5.84e6·38-s − 6.29e6·40-s + 9.05e6·41-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 0.209·4-s + 0.447·5-s − 0.631·7-s − 0.869·8-s + 0.491·10-s − 1.74·11-s + 1.15·13-s − 0.694·14-s − 1.16·16-s − 0.339·17-s − 0.413·19-s + 0.0936·20-s − 1.92·22-s − 1.74·23-s + 0.200·25-s + 1.27·26-s − 0.132·28-s + 0.121·29-s − 0.995·31-s − 0.412·32-s − 0.373·34-s − 0.282·35-s + 0.762·37-s − 0.454·38-s − 0.388·40-s + 0.500·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - 625T \) |
| good | 2 | \( 1 - 24.8T + 512T^{2} \) |
| 7 | \( 1 + 4.01e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.48e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.19e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.16e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.34e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.34e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.64e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.11e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.69e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 9.05e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 8.63e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.31e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.41e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.49e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.54e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.72e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.56e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.06e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.04e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.17e6T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.32e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.32e9T + 7.60e17T^{2} \) |
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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
−13.22507792103989415095560813895, −12.65656296573071284450520735566, −11.02586450709934580027793378523, −9.731498615145750304575527374568, −8.248680857667860900787177011590, −6.32824473247187077892180262011, −5.38467240505791407035913495882, −3.86224273573759658271626688990, −2.46626957077382352071554309710, 0,
2.46626957077382352071554309710, 3.86224273573759658271626688990, 5.38467240505791407035913495882, 6.32824473247187077892180262011, 8.248680857667860900787177011590, 9.731498615145750304575527374568, 11.02586450709934580027793378523, 12.65656296573071284450520735566, 13.22507792103989415095560813895
