L(s) = 1 | − 4·4-s − 4·7-s + 9-s − 6·11-s + 12·16-s + 18·23-s − 25-s + 16·28-s + 12·29-s − 4·36-s − 8·37-s − 14·43-s + 24·44-s + 9·49-s − 12·53-s − 4·63-s − 32·64-s − 8·67-s + 24·71-s + 24·77-s − 20·79-s + 81-s − 72·92-s − 6·99-s + 4·100-s + 18·107-s + 40·109-s + ⋯ |
L(s) = 1 | − 2·4-s − 1.51·7-s + 1/3·9-s − 1.80·11-s + 3·16-s + 3.75·23-s − 1/5·25-s + 3.02·28-s + 2.22·29-s − 2/3·36-s − 1.31·37-s − 2.13·43-s + 3.61·44-s + 9/7·49-s − 1.64·53-s − 0.503·63-s − 4·64-s − 0.977·67-s + 2.84·71-s + 2.73·77-s − 2.25·79-s + 1/9·81-s − 7.50·92-s − 0.603·99-s + 2/5·100-s + 1.74·107-s + 3.83·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 127449 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127449 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−9.139625451228424767056240959109, −8.618966621189813626434174913816, −8.522938517911553076706544737963, −7.76044884865200324165501143721, −7.22521844366863385910113943370, −6.66463575826783700250676263362, −6.13580656527827529767575928100, −5.14306131647545403565477997005, −5.04495215415329304235007921156, −4.76879607173120818228089465106, −3.73051779139000016834558062412, −3.01356233326132332503177574612, −3.00632648437536992753290178800, −1.06023509936170009525644393553, 0,
1.06023509936170009525644393553, 3.00632648437536992753290178800, 3.01356233326132332503177574612, 3.73051779139000016834558062412, 4.76879607173120818228089465106, 5.04495215415329304235007921156, 5.14306131647545403565477997005, 6.13580656527827529767575928100, 6.66463575826783700250676263362, 7.22521844366863385910113943370, 7.76044884865200324165501143721, 8.522938517911553076706544737963, 8.618966621189813626434174913816, 9.139625451228424767056240959109