| L(s) = 1 | − 2·2-s − 4-s − 4·7-s + 8·8-s + 3·9-s + 8·14-s − 7·16-s − 6·18-s + 14·23-s − 10·25-s + 4·28-s − 14·29-s − 14·32-s − 3·36-s + 10·37-s − 4·43-s − 28·46-s + 9·49-s + 20·50-s − 2·53-s − 32·56-s + 28·58-s − 12·63-s + 35·64-s − 24·67-s + 2·71-s + 24·72-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1/2·4-s − 1.51·7-s + 2.82·8-s + 9-s + 2.13·14-s − 7/4·16-s − 1.41·18-s + 2.91·23-s − 2·25-s + 0.755·28-s − 2.59·29-s − 2.47·32-s − 1/2·36-s + 1.64·37-s − 0.609·43-s − 4.12·46-s + 9/7·49-s + 2.82·50-s − 0.274·53-s − 4.27·56-s + 3.67·58-s − 1.51·63-s + 35/8·64-s − 2.93·67-s + 0.237·71-s + 2.82·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 53 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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, −8.860674573241203647498149622035, −8.308326610369561136757899186985, −7.54752000157688298165856122788, −7.21783238369309849566264053089, −7.20736947999270954805514319970, −6.04347889941024738587947019148, −5.74337021322869038057436488472, −4.77013937028831192931674609684, −4.50862835068296220400083952922, −3.68132447002767029896989401729, −3.31807614955039948665271389163, −1.98962412404329294063817388937, −1.08511071655936675723252049021, 0,
1.08511071655936675723252049021, 1.98962412404329294063817388937, 3.31807614955039948665271389163, 3.68132447002767029896989401729, 4.50862835068296220400083952922, 4.77013937028831192931674609684, 5.74337021322869038057436488472, 6.04347889941024738587947019148, 7.20736947999270954805514319970, 7.21783238369309849566264053089, 7.54752000157688298165856122788, 8.308326610369561136757899186985, 8.860674573241203647498149622035, 9.264238826193943461669666199356
