| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 9-s − 2·10-s + 2·12-s + 15-s − 4·16-s − 2·18-s + 2·20-s − 12·23-s + 25-s + 27-s − 10·29-s − 2·30-s + 8·32-s + 2·36-s + 8·43-s + 45-s + 24·46-s − 4·47-s − 4·48-s − 10·49-s − 2·50-s + 8·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1/3·9-s − 0.632·10-s + 0.577·12-s + 0.258·15-s − 16-s − 0.471·18-s + 0.447·20-s − 2.50·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 0.365·30-s + 1.41·32-s + 1/3·36-s + 1.21·43-s + 0.149·45-s + 3.53·46-s − 0.583·47-s − 0.577·48-s − 1.42·49-s − 0.282·50-s + 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( 1 - T \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.865210687516764254258578969287, −8.406166136751394246421595279767, −7.918932606252303436062518502827, −7.51918146688689824051279398861, −7.28038248877724312101019231600, −6.42434356455045288508739641500, −6.08163030557584203184343454820, −5.52528206592176090534176282599, −4.71505196722348200016560676535, −4.11721773489249008219589762002, −3.58877264279127956936756946841, −2.62944345863256205024284851048, −2.00208083660918387270836492277, −1.47809277983186343813078069607, 0,
1.47809277983186343813078069607, 2.00208083660918387270836492277, 2.62944345863256205024284851048, 3.58877264279127956936756946841, 4.11721773489249008219589762002, 4.71505196722348200016560676535, 5.52528206592176090534176282599, 6.08163030557584203184343454820, 6.42434356455045288508739641500, 7.28038248877724312101019231600, 7.51918146688689824051279398861, 7.918932606252303436062518502827, 8.406166136751394246421595279767, 8.865210687516764254258578969287
