| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 3·9-s − 2·11-s + 5·16-s + 6·18-s − 4·22-s − 16·23-s − 6·25-s − 10·29-s + 6·32-s + 9·36-s + 8·37-s − 16·43-s − 6·44-s − 32·46-s − 12·50-s − 12·53-s − 20·58-s + 7·64-s + 18·67-s − 4·71-s + 12·72-s + 16·74-s + 18·79-s − 32·86-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 9-s − 0.603·11-s + 5/4·16-s + 1.41·18-s − 0.852·22-s − 3.33·23-s − 6/5·25-s − 1.85·29-s + 1.06·32-s + 3/2·36-s + 1.31·37-s − 2.43·43-s − 0.904·44-s − 4.71·46-s − 1.69·50-s − 1.64·53-s − 2.62·58-s + 7/8·64-s + 2.19·67-s − 0.474·71-s + 1.41·72-s + 1.85·74-s + 2.02·79-s − 3.45·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−7.66334045549742507331716148152, −7.59001708095982935582996619965, −6.56153010353145194477530352450, −6.54204507036176471515809041147, −6.02860827782459493657287753528, −5.46355028016127947092103451217, −5.24362529829616318653741665156, −4.50301199161709359138099430647, −4.19558443542356955904809517882, −3.64499829703371390584081659599, −3.48832092996576829359255750084, −2.43990333774251370043130333163, −2.00011124513055881492662257488, −1.61277139597212650066132551280, 0,
1.61277139597212650066132551280, 2.00011124513055881492662257488, 2.43990333774251370043130333163, 3.48832092996576829359255750084, 3.64499829703371390584081659599, 4.19558443542356955904809517882, 4.50301199161709359138099430647, 5.24362529829616318653741665156, 5.46355028016127947092103451217, 6.02860827782459493657287753528, 6.54204507036176471515809041147, 6.56153010353145194477530352450, 7.59001708095982935582996619965, 7.66334045549742507331716148152
