| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 6·9-s − 2·10-s − 12·13-s + 16-s + 4·17-s − 6·18-s − 2·20-s + 3·25-s − 12·26-s + 12·29-s + 32-s + 4·34-s − 6·36-s − 20·37-s − 2·40-s + 4·41-s + 12·45-s + 49-s + 3·50-s − 12·52-s − 4·53-s + 12·58-s − 28·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 2·9-s − 0.632·10-s − 3.32·13-s + 1/4·16-s + 0.970·17-s − 1.41·18-s − 0.447·20-s + 3/5·25-s − 2.35·26-s + 2.22·29-s + 0.176·32-s + 0.685·34-s − 36-s − 3.28·37-s − 0.316·40-s + 0.624·41-s + 1.78·45-s + 1/7·49-s + 0.424·50-s − 1.66·52-s − 0.549·53-s + 1.57·58-s − 3.58·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.37599431086570880625729241811, −9.487980218157062241350495465657, −8.977646092372605556420891081568, −8.351664469300258231750592451668, −7.78536314488565228672834230956, −7.45420248729985818641828193491, −6.84363895294741728626832669651, −6.17616144092315208416665412995, −5.38123137825932555704959286891, −4.92587526788339694693667992483, −4.63923341949934829303628265619, −3.34536188954562334264634290221, −3.00381372509091040158022668795, −2.30801119480443055641880404256, 0,
2.30801119480443055641880404256, 3.00381372509091040158022668795, 3.34536188954562334264634290221, 4.63923341949934829303628265619, 4.92587526788339694693667992483, 5.38123137825932555704959286891, 6.17616144092315208416665412995, 6.84363895294741728626832669651, 7.45420248729985818641828193491, 7.78536314488565228672834230956, 8.351664469300258231750592451668, 8.977646092372605556420891081568, 9.487980218157062241350495465657, 10.37599431086570880625729241811
