| L(s) = 1 | + 3-s − 3·5-s + 7-s + 9-s − 11-s − 2·13-s − 3·15-s − 2·17-s + 21-s − 8·23-s + 5·25-s + 27-s − 4·29-s − 5·31-s − 33-s − 3·35-s − 6·37-s − 2·39-s − 6·41-s + 4·43-s − 3·45-s − 8·47-s − 5·49-s − 2·51-s + 11·53-s + 3·55-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.774·15-s − 0.485·17-s + 0.218·21-s − 1.66·23-s + 25-s + 0.192·27-s − 0.742·29-s − 0.898·31-s − 0.174·33-s − 0.507·35-s − 0.986·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.447·45-s − 1.16·47-s − 5/7·49-s − 0.280·51-s + 1.51·53-s + 0.404·55-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 46 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 112 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 4 T - 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 68 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 7 T + 58 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 + 3 T - 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−14.5628775049, −14.0855242694, −13.6073307333, −13.1385205616, −12.6524420319, −12.2057921416, −11.7874154785, −11.4692781350, −10.8836051245, −10.4667885263, −9.92113203800, −9.43873728918, −8.80741751959, −8.40390660446, −7.90550864717, −7.64863383820, −7.03961338024, −6.60990731223, −5.72007337484, −5.10010829427, −4.54506754421, −3.77662824686, −3.60335237189, −2.52861083897, −1.79201203511, 0,
1.79201203511, 2.52861083897, 3.60335237189, 3.77662824686, 4.54506754421, 5.10010829427, 5.72007337484, 6.60990731223, 7.03961338024, 7.64863383820, 7.90550864717, 8.40390660446, 8.80741751959, 9.43873728918, 9.92113203800, 10.4667885263, 10.8836051245, 11.4692781350, 11.7874154785, 12.2057921416, 12.6524420319, 13.1385205616, 13.6073307333, 14.0855242694, 14.5628775049
