| L(s) = 1 | + 2·2-s + 2·4-s + 2·7-s + 9-s + 4·14-s − 4·16-s + 2·18-s + 8·23-s − 6·25-s + 4·28-s − 8·32-s + 2·36-s + 16·46-s − 3·49-s − 12·50-s + 2·63-s − 8·64-s − 24·71-s − 20·79-s + 81-s + 16·92-s − 6·98-s − 12·100-s − 8·112-s + 12·113-s + 22·121-s + 4·126-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.755·7-s + 1/3·9-s + 1.06·14-s − 16-s + 0.471·18-s + 1.66·23-s − 6/5·25-s + 0.755·28-s − 1.41·32-s + 1/3·36-s + 2.35·46-s − 3/7·49-s − 1.69·50-s + 0.251·63-s − 64-s − 2.84·71-s − 2.25·79-s + 1/9·81-s + 1.66·92-s − 0.606·98-s − 6/5·100-s − 0.755·112-s + 1.12·113-s + 2·121-s + 0.356·126-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.600289635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.600289635\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.88227645139139147375234207265, −10.07425476101980981778187468902, −9.635512448758663864614166679485, −8.796837588933868122878048851201, −8.594881069987249738386887227349, −7.59016702588762759857160629806, −7.28071169368978645204293409172, −6.58378473240236249328576328281, −5.87632723717406078403508590830, −5.44049680085649235006088097063, −4.65163974920147130957062766615, −4.38916726017069099428819447218, −3.47485328369714008948838554266, −2.78364360095173076713014342455, −1.68594557842079147034482742426,
1.68594557842079147034482742426, 2.78364360095173076713014342455, 3.47485328369714008948838554266, 4.38916726017069099428819447218, 4.65163974920147130957062766615, 5.44049680085649235006088097063, 5.87632723717406078403508590830, 6.58378473240236249328576328281, 7.28071169368978645204293409172, 7.59016702588762759857160629806, 8.594881069987249738386887227349, 8.796837588933868122878048851201, 9.635512448758663864614166679485, 10.07425476101980981778187468902, 10.88227645139139147375234207265
