| L(s) = 1 | − 2·3-s − 4·7-s + 2·9-s − 2·11-s + 2·13-s + 6·19-s + 8·21-s − 12·23-s − 6·27-s − 6·29-s + 16·31-s + 4·33-s − 6·37-s − 4·39-s − 10·43-s − 2·49-s − 10·53-s − 12·57-s − 6·59-s − 18·61-s − 8·63-s + 10·67-s + 24·69-s − 8·73-s + 8·77-s + 11·81-s − 2·83-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s + 2/3·9-s − 0.603·11-s + 0.554·13-s + 1.37·19-s + 1.74·21-s − 2.50·23-s − 1.15·27-s − 1.11·29-s + 2.87·31-s + 0.696·33-s − 0.986·37-s − 0.640·39-s − 1.52·43-s − 2/7·49-s − 1.37·53-s − 1.58·57-s − 0.781·59-s − 2.30·61-s − 1.00·63-s + 1.22·67-s + 2.88·69-s − 0.936·73-s + 0.911·77-s + 11/9·81-s − 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.409849107021290924212913667598, −8.973240765902173635808131042840, −8.166221422442874923142527804299, −8.021975407540844688834874989070, −7.69637422811741651850934558385, −7.10033155884986463762739968618, −6.47840116003058930629506805179, −6.42877478120596479587860185234, −5.87631680476208146052773780967, −5.82865531445608805589229308253, −5.00401420807203998386668496800, −4.86788396244498416072826967746, −4.09648511318576568001315043004, −3.66254060472490055928832640638, −3.18816530161154160413273479350, −2.77741016294620158090345512097, −1.87310610378703042054930399652, −1.28971217048944487230143332373, 0, 0,
1.28971217048944487230143332373, 1.87310610378703042054930399652, 2.77741016294620158090345512097, 3.18816530161154160413273479350, 3.66254060472490055928832640638, 4.09648511318576568001315043004, 4.86788396244498416072826967746, 5.00401420807203998386668496800, 5.82865531445608805589229308253, 5.87631680476208146052773780967, 6.42877478120596479587860185234, 6.47840116003058930629506805179, 7.10033155884986463762739968618, 7.69637422811741651850934558385, 8.021975407540844688834874989070, 8.166221422442874923142527804299, 8.973240765902173635808131042840, 9.409849107021290924212913667598
