| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s − 2·9-s + 2·12-s − 4·16-s − 4·18-s − 5·19-s − 8·23-s − 5·27-s − 8·32-s − 4·36-s − 10·38-s − 8·43-s − 16·46-s + 4·47-s − 4·48-s + 5·49-s − 8·53-s − 10·54-s − 5·57-s − 8·64-s − 67-s − 8·69-s + 4·71-s − 8·73-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s − 2/3·9-s + 0.577·12-s − 16-s − 0.942·18-s − 1.14·19-s − 1.66·23-s − 0.962·27-s − 1.41·32-s − 2/3·36-s − 1.62·38-s − 1.21·43-s − 2.35·46-s + 0.583·47-s − 0.577·48-s + 5/7·49-s − 1.09·53-s − 1.36·54-s − 0.662·57-s − 64-s − 0.122·67-s − 0.963·69-s + 0.474·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−8.447619922078198335972158347179, −8.044827370713296422123332780690, −7.59362090901096340607979088197, −6.84634214489735559691502009567, −6.52344252442029914050091017131, −5.92239847994350914236194755762, −5.69115475967100998513405578845, −5.08841001626034087244742493296, −4.40626726910186118384036177680, −4.10376009888692646810309680571, −3.50627961148140796102205348950, −2.98839093361595408735780019351, −2.33967136269074927880433226336, −1.83094158162311263720336336452, 0,
1.83094158162311263720336336452, 2.33967136269074927880433226336, 2.98839093361595408735780019351, 3.50627961148140796102205348950, 4.10376009888692646810309680571, 4.40626726910186118384036177680, 5.08841001626034087244742493296, 5.69115475967100998513405578845, 5.92239847994350914236194755762, 6.52344252442029914050091017131, 6.84634214489735559691502009567, 7.59362090901096340607979088197, 8.044827370713296422123332780690, 8.447619922078198335972158347179
