| L(s) = 1 | − 2·3-s − 2·5-s − 2·7-s + 2·9-s − 8·11-s − 6·13-s + 4·15-s − 6·17-s + 4·21-s − 6·23-s − 25-s − 6·27-s + 4·29-s + 16·33-s + 4·35-s − 6·37-s + 12·39-s + 12·41-s + 6·43-s − 4·45-s − 18·47-s + 2·49-s + 12·51-s + 10·53-s + 16·55-s − 4·63-s + 12·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 0.755·7-s + 2/3·9-s − 2.41·11-s − 1.66·13-s + 1.03·15-s − 1.45·17-s + 0.872·21-s − 1.25·23-s − 1/5·25-s − 1.15·27-s + 0.742·29-s + 2.78·33-s + 0.676·35-s − 0.986·37-s + 1.92·39-s + 1.87·41-s + 0.914·43-s − 0.596·45-s − 2.62·47-s + 2/7·49-s + 1.68·51-s + 1.37·53-s + 2.15·55-s − 0.503·63-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−11.18185980305712437639796722003, −11.16334154463584819392357277984, −10.42895204673257971173588717182, −10.24952265206505591234630510982, −9.549039434478250297792296135029, −9.387384252825253987340778726341, −8.228006242855630454386827497975, −8.029120002825090631820258281723, −7.63947010131915700151856943361, −6.93738955347480394716968290260, −6.64235707921365981905084689582, −5.90140119677323670090460866986, −5.19357968208382330038713759751, −5.14948456479960198295978311814, −4.28202416829487687819221214242, −3.79198998007098028885591972392, −2.57119187761471274617756645711, −2.39267600089662764382802844966, 0, 0,
2.39267600089662764382802844966, 2.57119187761471274617756645711, 3.79198998007098028885591972392, 4.28202416829487687819221214242, 5.14948456479960198295978311814, 5.19357968208382330038713759751, 5.90140119677323670090460866986, 6.64235707921365981905084689582, 6.93738955347480394716968290260, 7.63947010131915700151856943361, 8.029120002825090631820258281723, 8.228006242855630454386827497975, 9.387384252825253987340778726341, 9.549039434478250297792296135029, 10.24952265206505591234630510982, 10.42895204673257971173588717182, 11.16334154463584819392357277984, 11.18185980305712437639796722003
