| L(s) = 1 | − 2·2-s + 2·4-s − 4·5-s − 6·9-s + 8·10-s − 4·13-s − 4·16-s − 16·17-s + 12·18-s − 8·20-s + 11·25-s + 8·26-s − 8·29-s + 8·32-s + 32·34-s − 12·36-s − 10·37-s + 18·41-s + 24·45-s − 22·50-s − 8·52-s + 16·58-s − 20·61-s − 8·64-s + 16·65-s − 32·68-s + 22·73-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.78·5-s − 2·9-s + 2.52·10-s − 1.10·13-s − 16-s − 3.88·17-s + 2.82·18-s − 1.78·20-s + 11/5·25-s + 1.56·26-s − 1.48·29-s + 1.41·32-s + 5.48·34-s − 2·36-s − 1.64·37-s + 2.81·41-s + 3.57·45-s − 3.11·50-s − 1.10·52-s + 2.10·58-s − 2.56·61-s − 64-s + 1.98·65-s − 3.88·68-s + 2.57·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−11.41969885717196073800069890779, −11.09640815561493830712783322151, −10.94359038151489106072752521043, −10.68282862682245510106836463429, −9.490732212704087439628064325530, −9.173000378189406264853118695295, −8.796479618661795785853957972221, −8.565905979708743877436971470991, −7.75805147507824238498795753363, −7.72064250284630075532398206771, −6.82573504118796423410429596177, −6.69126558112912186218081072837, −5.74634980679457322559189331981, −4.82913331306829617973868572676, −4.44232462347619551420660772294, −3.74508564049840845926115165695, −2.65457487424769336016037753336, −2.25357395187725162810728799257, 0, 0,
2.25357395187725162810728799257, 2.65457487424769336016037753336, 3.74508564049840845926115165695, 4.44232462347619551420660772294, 4.82913331306829617973868572676, 5.74634980679457322559189331981, 6.69126558112912186218081072837, 6.82573504118796423410429596177, 7.72064250284630075532398206771, 7.75805147507824238498795753363, 8.565905979708743877436971470991, 8.796479618661795785853957972221, 9.173000378189406264853118695295, 9.490732212704087439628064325530, 10.68282862682245510106836463429, 10.94359038151489106072752521043, 11.09640815561493830712783322151, 11.41969885717196073800069890779
