| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 10-s − 2·13-s − 16-s + 6·17-s − 4·19-s − 20-s − 8·23-s + 25-s + 2·26-s + 6·29-s − 8·31-s − 5·32-s − 6·34-s + 6·37-s + 4·38-s + 3·40-s + 6·41-s + 4·43-s + 8·46-s − 8·47-s − 50-s + 2·52-s + 10·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s − 0.554·13-s − 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.223·20-s − 1.66·23-s + 1/5·25-s + 0.392·26-s + 1.11·29-s − 1.43·31-s − 0.883·32-s − 1.02·34-s + 0.986·37-s + 0.648·38-s + 0.474·40-s + 0.937·41-s + 0.609·43-s + 1.17·46-s − 1.16·47-s − 0.141·50-s + 0.277·52-s + 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.96763757306948, −12.49391299418908, −12.22234008998787, −11.68727206252209, −10.93458137149678, −10.55970586032478, −10.17930372846236, −9.786858060114677, −9.300520278352305, −9.017976531618949, −8.271892019037389, −8.010774218399384, −7.515054468878086, −7.153948087364358, −6.268254580619123, −5.950231155242019, −5.490998069535740, −4.782011324095820, −4.401259875350390, −3.890604446140559, −3.217895208393209, −2.566604466307857, −1.896364128454368, −1.431586139932085, −0.6812247555787184, 0,
0.6812247555787184, 1.431586139932085, 1.896364128454368, 2.566604466307857, 3.217895208393209, 3.890604446140559, 4.401259875350390, 4.782011324095820, 5.490998069535740, 5.950231155242019, 6.268254580619123, 7.153948087364358, 7.515054468878086, 8.010774218399384, 8.271892019037389, 9.017976531618949, 9.300520278352305, 9.786858060114677, 10.17930372846236, 10.55970586032478, 10.93458137149678, 11.68727206252209, 12.22234008998787, 12.49391299418908, 12.96763757306948
