| L(s) = 1 | + 3-s − 4·4-s + 4·7-s + 9-s − 4·12-s − 2·13-s + 12·16-s + 2·19-s + 4·21-s − 10·25-s + 27-s − 16·28-s + 16·31-s − 4·36-s + 4·37-s − 2·39-s + 16·43-s + 12·48-s − 2·49-s + 8·52-s + 2·57-s − 20·61-s + 4·63-s − 32·64-s − 20·67-s − 8·73-s − 10·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2·4-s + 1.51·7-s + 1/3·9-s − 1.15·12-s − 0.554·13-s + 3·16-s + 0.458·19-s + 0.872·21-s − 2·25-s + 0.192·27-s − 3.02·28-s + 2.87·31-s − 2/3·36-s + 0.657·37-s − 0.320·39-s + 2.43·43-s + 1.73·48-s − 2/7·49-s + 1.10·52-s + 0.264·57-s − 2.56·61-s + 0.503·63-s − 4·64-s − 2.44·67-s − 0.936·73-s − 1.15·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1179387 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1179387 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 - T \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.047058109228960310569118011279, −7.57873319845860197697619507154, −7.38987844970363499694447516693, −6.28975454567733783999194978856, −5.92615105889325146373690957476, −5.51875395795692112491347722200, −4.88303727227954019406472250536, −4.58543437562245729394210684131, −4.12480866415710071026089413746, −4.10497206474885287550986629629, −2.89395743647375258219896560652, −2.81428608433619659458677251434, −1.55429192170819761115384444567, −1.21678682746529038884045056230, 0,
1.21678682746529038884045056230, 1.55429192170819761115384444567, 2.81428608433619659458677251434, 2.89395743647375258219896560652, 4.10497206474885287550986629629, 4.12480866415710071026089413746, 4.58543437562245729394210684131, 4.88303727227954019406472250536, 5.51875395795692112491347722200, 5.92615105889325146373690957476, 6.28975454567733783999194978856, 7.38987844970363499694447516693, 7.57873319845860197697619507154, 8.047058109228960310569118011279
