| L(s) = 1 | − 4-s + 4·7-s − 2·13-s − 3·16-s + 4·19-s − 7·25-s − 4·28-s + 16·31-s − 14·37-s + 4·43-s − 2·49-s + 2·52-s − 14·61-s + 7·64-s − 20·67-s − 14·73-s − 4·76-s + 4·79-s − 8·91-s + 4·97-s + 7·100-s + 16·103-s + 22·109-s − 12·112-s − 10·121-s − 16·124-s + 127-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.51·7-s − 0.554·13-s − 3/4·16-s + 0.917·19-s − 7/5·25-s − 0.755·28-s + 2.87·31-s − 2.30·37-s + 0.609·43-s − 2/7·49-s + 0.277·52-s − 1.79·61-s + 7/8·64-s − 2.44·67-s − 1.63·73-s − 0.458·76-s + 0.450·79-s − 0.838·91-s + 0.406·97-s + 7/10·100-s + 1.57·103-s + 2.10·109-s − 1.13·112-s − 0.909·121-s − 1.43·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6561 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8950357202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8950357202\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−14.29964332369256398717487679596, −14.24534143580850222689806450921, −13.53005878760745684919879538913, −13.39369631200027704092706736493, −12.22520072356073210718671997465, −11.88045000978665525853298817583, −11.60281525157102031033473031367, −10.81679773430834793125056262951, −10.20242410709674926896312591641, −9.699851402202700459913494023810, −8.922789416667976924339583277092, −8.462010452150115627851294225307, −7.76833571382616952776524821959, −7.36311860625225098919703816696, −6.38667701978104996146814405770, −5.56324876902780652194785160059, −4.60289706621031782825417838661, −4.58603454223708864059824147194, −3.12234009529020877746547298161, −1.78651121087012653708241175100,
1.78651121087012653708241175100, 3.12234009529020877746547298161, 4.58603454223708864059824147194, 4.60289706621031782825417838661, 5.56324876902780652194785160059, 6.38667701978104996146814405770, 7.36311860625225098919703816696, 7.76833571382616952776524821959, 8.462010452150115627851294225307, 8.922789416667976924339583277092, 9.699851402202700459913494023810, 10.20242410709674926896312591641, 10.81679773430834793125056262951, 11.60281525157102031033473031367, 11.88045000978665525853298817583, 12.22520072356073210718671997465, 13.39369631200027704092706736493, 13.53005878760745684919879538913, 14.24534143580850222689806450921, 14.29964332369256398717487679596
