| L(s) = 1 | − 2-s − 4-s − 8·7-s + 3·8-s + 3·9-s + 8·14-s − 16-s − 6·17-s − 3·18-s + 14·23-s − 10·25-s + 8·28-s + 8·31-s − 5·32-s + 6·34-s − 3·36-s + 12·41-s − 14·46-s − 4·47-s + 34·49-s + 10·50-s − 24·56-s − 8·62-s − 24·63-s + 7·64-s + 6·68-s + 2·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 3.02·7-s + 1.06·8-s + 9-s + 2.13·14-s − 1/4·16-s − 1.45·17-s − 0.707·18-s + 2.91·23-s − 2·25-s + 1.51·28-s + 1.43·31-s − 0.883·32-s + 1.02·34-s − 1/2·36-s + 1.87·41-s − 2.06·46-s − 0.583·47-s + 34/7·49-s + 1.41·50-s − 3.20·56-s − 1.01·62-s − 3.02·63-s + 7/8·64-s + 0.727·68-s + 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 179776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + p T^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.264238826193943461669666199356, −8.630974269277868498897922665671, −8.084989976983600985292488466306, −7.20736947999270954805514319970, −7.06049552371860101957180575055, −6.72056035585969494914632376462, −6.04347889941024738587947019148, −5.67860390961286302579935116669, −4.50862835068296220400083952922, −4.44617145148605633628123917525, −3.59167852485893419100627631769, −3.08557333060438750476516341512, −2.38251275454171592523856823732, −1.03429417110588513889366691339, 0,
1.03429417110588513889366691339, 2.38251275454171592523856823732, 3.08557333060438750476516341512, 3.59167852485893419100627631769, 4.44617145148605633628123917525, 4.50862835068296220400083952922, 5.67860390961286302579935116669, 6.04347889941024738587947019148, 6.72056035585969494914632376462, 7.06049552371860101957180575055, 7.20736947999270954805514319970, 8.084989976983600985292488466306, 8.630974269277868498897922665671, 9.264238826193943461669666199356
