L(s) = 1 | − 4·2-s − 3-s + 8·4-s + 4·6-s − 4·7-s − 8·8-s − 2·9-s − 8·12-s + 16·14-s − 4·16-s + 8·18-s + 4·21-s + 8·24-s − 9·25-s + 5·27-s − 32·28-s + 32·32-s − 16·36-s − 16·41-s − 16·42-s − 12·43-s + 4·48-s − 2·49-s + 36·50-s − 12·53-s − 20·54-s + 32·56-s + ⋯ |
L(s) = 1 | − 2.82·2-s − 0.577·3-s + 4·4-s + 1.63·6-s − 1.51·7-s − 2.82·8-s − 2/3·9-s − 2.30·12-s + 4.27·14-s − 16-s + 1.88·18-s + 0.872·21-s + 1.63·24-s − 9/5·25-s + 0.962·27-s − 6.04·28-s + 5.65·32-s − 8/3·36-s − 2.49·41-s − 2.46·42-s − 1.82·43-s + 0.577·48-s − 2/7·49-s + 5.09·50-s − 1.64·53-s − 2.72·54-s + 4.27·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 393129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.603539619290756001226038948684, −8.214596534850172998316450386324, −7.73556351389179885223258451389, −7.17115755169439685292874478446, −6.77654603285244896156727941052, −6.36261389471308870138602900888, −6.05894787659039661644210218582, −5.09690794065788751110696140894, −4.80674258105858635103949879256, −3.62924072940666698491701863630, −3.33280490374154022682414025774, −2.24326475114330249154243176400, −1.78787687456595466327825076524, −0.62260408018451229277893602459, 0,
0.62260408018451229277893602459, 1.78787687456595466327825076524, 2.24326475114330249154243176400, 3.33280490374154022682414025774, 3.62924072940666698491701863630, 4.80674258105858635103949879256, 5.09690794065788751110696140894, 6.05894787659039661644210218582, 6.36261389471308870138602900888, 6.77654603285244896156727941052, 7.17115755169439685292874478446, 7.73556351389179885223258451389, 8.214596534850172998316450386324, 8.603539619290756001226038948684