| L(s) = 1 | − 8·17-s + 8·23-s − 16·31-s + 16·41-s − 8·47-s − 6·49-s + 8·71-s − 16·73-s − 16·79-s + 8·89-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s + ⋯ |
| L(s) = 1 | − 1.94·17-s + 1.66·23-s − 2.87·31-s + 2.49·41-s − 1.16·47-s − 6/7·49-s + 0.949·71-s − 1.87·73-s − 1.80·79-s + 0.847·89-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.181·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6/13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 16 T + 118 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 16 T + 144 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 192 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 226 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−13.1656858696, −12.9365685309, −12.5951339099, −11.8497766649, −11.4090311903, −11.1110399443, −10.8097671692, −10.4931377813, −9.68654370857, −9.33001168649, −8.97473122114, −8.74403503609, −8.07953861831, −7.45559021064, −7.19154556749, −6.75143305821, −6.14135545604, −5.74972234835, −5.01918327493, −4.71625850807, −4.05302007325, −3.52813734470, −2.77184080503, −2.19096904361, −1.36916312040, 0,
1.36916312040, 2.19096904361, 2.77184080503, 3.52813734470, 4.05302007325, 4.71625850807, 5.01918327493, 5.74972234835, 6.14135545604, 6.75143305821, 7.19154556749, 7.45559021064, 8.07953861831, 8.74403503609, 8.97473122114, 9.33001168649, 9.68654370857, 10.4931377813, 10.8097671692, 11.1110399443, 11.4090311903, 11.8497766649, 12.5951339099, 12.9365685309, 13.1656858696
