L(s) = 1 | + 2-s + 4-s − 7-s + 8-s + 2·9-s − 14-s + 16-s + 2·18-s − 8·23-s − 6·25-s − 28-s + 4·29-s + 32-s + 2·36-s + 4·37-s − 8·46-s + 49-s − 6·50-s − 4·53-s − 56-s + 4·58-s − 2·63-s + 64-s − 24·67-s + 16·71-s + 2·72-s + 4·74-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 2/3·9-s − 0.267·14-s + 1/4·16-s + 0.471·18-s − 1.66·23-s − 6/5·25-s − 0.188·28-s + 0.742·29-s + 0.176·32-s + 1/3·36-s + 0.657·37-s − 1.17·46-s + 1/7·49-s − 0.848·50-s − 0.549·53-s − 0.133·56-s + 0.525·58-s − 0.251·63-s + 1/8·64-s − 2.93·67-s + 1.89·71-s + 0.235·72-s + 0.464·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.479234028\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.479234028\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( 1 + T \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
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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
−11.67243765067819893667519287194, −10.86030205769484123590609978555, −10.38154541301378339571912207256, −9.780964096385823605224897000939, −9.440497965278766401080888076377, −8.466149797686215063234877480194, −7.87080399329017938577900963261, −7.37229758267953838614619646206, −6.54361952531080618560362273781, −6.09461834558935249753190776977, −5.41783473797801293529811804623, −4.41783868169106956331622880408, −4.00514677824647069794495389872, −3.00486518899045402623792404204, −1.88056630179972506147475318583,
1.88056630179972506147475318583, 3.00486518899045402623792404204, 4.00514677824647069794495389872, 4.41783868169106956331622880408, 5.41783473797801293529811804623, 6.09461834558935249753190776977, 6.54361952531080618560362273781, 7.37229758267953838614619646206, 7.87080399329017938577900963261, 8.466149797686215063234877480194, 9.440497965278766401080888076377, 9.780964096385823605224897000939, 10.38154541301378339571912207256, 10.86030205769484123590609978555, 11.67243765067819893667519287194