| L(s) = 1 | + 3-s + 4-s + 9-s + 12-s − 12·13-s + 16-s + 8·19-s − 6·25-s + 27-s + 36-s − 20·37-s − 12·39-s − 8·43-s + 48-s − 12·52-s + 8·57-s − 12·61-s + 64-s + 8·67-s − 20·73-s − 6·75-s + 8·76-s + 81-s + 28·97-s − 6·100-s − 16·103-s + 108-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 1/3·9-s + 0.288·12-s − 3.32·13-s + 1/4·16-s + 1.83·19-s − 6/5·25-s + 0.192·27-s + 1/6·36-s − 3.28·37-s − 1.92·39-s − 1.21·43-s + 0.144·48-s − 1.66·52-s + 1.05·57-s − 1.53·61-s + 1/8·64-s + 0.977·67-s − 2.34·73-s − 0.692·75-s + 0.917·76-s + 1/9·81-s + 2.84·97-s − 3/5·100-s − 1.57·103-s + 0.0962·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259308 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259308 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−8.882543700432626086795887126363, −7.954464142207805330736993023418, −7.68532191797102726979476428936, −7.36399840290663473896741338506, −6.95470194591461271128583103192, −6.54855688605176886120203881032, −5.47990216984766844713395370376, −5.32455191890654195761811472253, −4.82857955948335988380640999352, −4.16943092436514571048096348777, −3.17941538941685381741709304651, −3.08481257245522303063597583848, −2.16063348217202322480816158281, −1.71238689353652455033144003099, 0,
1.71238689353652455033144003099, 2.16063348217202322480816158281, 3.08481257245522303063597583848, 3.17941538941685381741709304651, 4.16943092436514571048096348777, 4.82857955948335988380640999352, 5.32455191890654195761811472253, 5.47990216984766844713395370376, 6.54855688605176886120203881032, 6.95470194591461271128583103192, 7.36399840290663473896741338506, 7.68532191797102726979476428936, 7.954464142207805330736993023418, 8.882543700432626086795887126363
