| L(s) = 1 | − 8·7-s + 3·9-s + 4·11-s − 10·13-s − 4·19-s + 23-s − 6·25-s − 14·29-s − 18·41-s − 16·43-s + 34·49-s − 24·63-s + 28·67-s − 6·73-s − 32·77-s − 12·79-s + 16·83-s + 80·91-s + 12·99-s + 4·101-s − 16·103-s + 4·107-s − 30·117-s − 10·121-s + 127-s + 131-s + 32·133-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 9-s + 1.20·11-s − 2.77·13-s − 0.917·19-s + 0.208·23-s − 6/5·25-s − 2.59·29-s − 2.81·41-s − 2.43·43-s + 34/7·49-s − 3.02·63-s + 3.42·67-s − 0.702·73-s − 3.64·77-s − 1.35·79-s + 1.75·83-s + 8.38·91-s + 1.20·99-s + 0.398·101-s − 1.57·103-s + 0.386·107-s − 2.77·117-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 778688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 778688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 23 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−7.41247065200411840033758437727, −7.39242312760941532932793610626, −6.76522353190121364379140827555, −6.53483233561451653204235369889, −6.41753135284542253331461685383, −5.32486349366702261701788999466, −5.32154529144374630720020311491, −4.41717411841994123441247215854, −3.80206517892802911800653447979, −3.59326225075188741908876070363, −3.06248996148821975099005271438, −2.22648022622692068420989343553, −1.80022870583675220558113437114, 0, 0,
1.80022870583675220558113437114, 2.22648022622692068420989343553, 3.06248996148821975099005271438, 3.59326225075188741908876070363, 3.80206517892802911800653447979, 4.41717411841994123441247215854, 5.32154529144374630720020311491, 5.32486349366702261701788999466, 6.41753135284542253331461685383, 6.53483233561451653204235369889, 6.76522353190121364379140827555, 7.39242312760941532932793610626, 7.41247065200411840033758437727
