| L(s) = 1 | + 2-s + 3-s + 5-s + 6-s − 8-s + 9-s + 10-s + 15-s − 16-s + 18-s − 23-s − 24-s + 2·27-s − 2·29-s + 30-s − 40-s + 2·41-s + 2·43-s + 45-s − 46-s − 2·47-s − 48-s + 2·54-s − 2·58-s − 61-s + 64-s − 67-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 5-s + 6-s − 8-s + 9-s + 10-s + 15-s − 16-s + 18-s − 23-s − 24-s + 2·27-s − 2·29-s + 30-s − 40-s + 2·41-s + 2·43-s + 45-s − 46-s − 2·47-s − 48-s + 2·54-s − 2·58-s − 61-s + 64-s − 67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.305322459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.305322459\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−10.27063485513183203739136769420, −9.927437613709668642929250081885, −9.367407458892367353981578338808, −9.306118320026644146129656688557, −8.920743982118600142009791349902, −8.386841777392394641761873334448, −7.78613230860870997643551956293, −7.61834630716422648512255728174, −6.99710191533280619868720604093, −6.28411092865483912116435593223, −6.23179800747396393729281022283, −5.43849516585984197964514009485, −5.42034426440017055741925048047, −4.50244504137645818918700393510, −4.17212403703495731685669257476, −3.86209325741397357243344498892, −3.01324797728851002423687673662, −2.75032645736142701288441007675, −2.07868888227772951163696460029, −1.44552514691148156794630113023,
1.44552514691148156794630113023, 2.07868888227772951163696460029, 2.75032645736142701288441007675, 3.01324797728851002423687673662, 3.86209325741397357243344498892, 4.17212403703495731685669257476, 4.50244504137645818918700393510, 5.42034426440017055741925048047, 5.43849516585984197964514009485, 6.23179800747396393729281022283, 6.28411092865483912116435593223, 6.99710191533280619868720604093, 7.61834630716422648512255728174, 7.78613230860870997643551956293, 8.386841777392394641761873334448, 8.920743982118600142009791349902, 9.306118320026644146129656688557, 9.367407458892367353981578338808, 9.927437613709668642929250081885, 10.27063485513183203739136769420
