| L(s) = 1 | + 2·2-s + 3·4-s − 4·7-s + 4·8-s − 9-s − 4·11-s − 8·14-s + 5·16-s − 2·18-s − 8·22-s − 6·23-s − 8·25-s − 12·28-s + 8·29-s + 6·32-s − 3·36-s − 4·37-s + 2·43-s − 12·44-s − 12·46-s + 9·49-s − 16·50-s − 26·53-s − 16·56-s + 16·58-s + 4·63-s + 7·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.51·7-s + 1.41·8-s − 1/3·9-s − 1.20·11-s − 2.13·14-s + 5/4·16-s − 0.471·18-s − 1.70·22-s − 1.25·23-s − 8/5·25-s − 2.26·28-s + 1.48·29-s + 1.06·32-s − 1/2·36-s − 0.657·37-s + 0.304·43-s − 1.80·44-s − 1.76·46-s + 9/7·49-s − 2.26·50-s − 3.57·53-s − 2.13·56-s + 2.10·58-s + 0.503·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
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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
−8.746596566655627372227816500830, −8.061198584785749060904406475426, −7.85694989382193453068773222919, −7.27621615658792238413137504693, −6.52942432980064213361944418884, −6.32909157472471250011099205825, −5.90254046530571968266193465680, −5.34163701086578957197840051403, −4.84889969602696966910786363246, −4.15551808462629396511001214996, −3.64945365799186774562776746606, −3.02452322121104901260433748870, −2.65219906650047338477008693772, −1.82688945775412796314629214483, 0,
1.82688945775412796314629214483, 2.65219906650047338477008693772, 3.02452322121104901260433748870, 3.64945365799186774562776746606, 4.15551808462629396511001214996, 4.84889969602696966910786363246, 5.34163701086578957197840051403, 5.90254046530571968266193465680, 6.32909157472471250011099205825, 6.52942432980064213361944418884, 7.27621615658792238413137504693, 7.85694989382193453068773222919, 8.061198584785749060904406475426, 8.746596566655627372227816500830
