| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s − 5·9-s − 4·14-s + 16-s + 6·17-s − 5·18-s − 12·23-s − 4·28-s + 4·31-s + 32-s + 6·34-s − 5·36-s − 6·41-s − 12·46-s − 24·47-s − 2·49-s − 4·56-s + 4·62-s + 20·63-s + 64-s + 6·68-s + 24·71-s − 5·72-s − 22·73-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s − 5/3·9-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 1.17·18-s − 2.50·23-s − 0.755·28-s + 0.718·31-s + 0.176·32-s + 1.02·34-s − 5/6·36-s − 0.937·41-s − 1.76·46-s − 3.50·47-s − 2/7·49-s − 0.534·56-s + 0.508·62-s + 2.51·63-s + 1/8·64-s + 0.727·68-s + 2.84·71-s − 0.589·72-s − 2.57·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 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
−9.828211755289408046473885177280, −8.984879988992018021562140214035, −8.305108042202074861058072254432, −8.071729218562685355881418106440, −7.50685211643047093775514357787, −6.51600833722009744085233889545, −6.35306147256574072432179047412, −5.92775363035311842870660338443, −5.34945783064988058695168525193, −4.74379980896905468893340943589, −3.73692800680606925562186179410, −3.29699975126484092788987476482, −2.93297962179085959554069000009, −1.90088015093802268091220378336, 0,
1.90088015093802268091220378336, 2.93297962179085959554069000009, 3.29699975126484092788987476482, 3.73692800680606925562186179410, 4.74379980896905468893340943589, 5.34945783064988058695168525193, 5.92775363035311842870660338443, 6.35306147256574072432179047412, 6.51600833722009744085233889545, 7.50685211643047093775514357787, 8.071729218562685355881418106440, 8.305108042202074861058072254432, 8.984879988992018021562140214035, 9.828211755289408046473885177280
