| L(s) = 1 | + 3-s − 8·7-s + 9-s − 4·13-s − 8·19-s − 8·21-s − 6·25-s + 27-s + 8·31-s − 4·37-s − 4·39-s + 8·43-s + 34·49-s − 8·57-s + 12·61-s − 8·63-s + 8·67-s − 12·73-s − 6·75-s + 8·79-s + 81-s + 32·91-s + 8·93-s − 28·97-s − 24·103-s + 28·109-s − 4·111-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 3.02·7-s + 1/3·9-s − 1.10·13-s − 1.83·19-s − 1.74·21-s − 6/5·25-s + 0.192·27-s + 1.43·31-s − 0.657·37-s − 0.640·39-s + 1.21·43-s + 34/7·49-s − 1.05·57-s + 1.53·61-s − 1.00·63-s + 0.977·67-s − 1.40·73-s − 0.692·75-s + 0.900·79-s + 1/9·81-s + 3.35·91-s + 0.829·93-s − 2.84·97-s − 2.36·103-s + 2.68·109-s − 0.379·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27648 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27648 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−10.23151567133484212790849208076, −9.689914794558097820657325082483, −9.488699237661810053920246470193, −8.839886292016704738441339481672, −8.291996087541312922253124488251, −7.52219997040636286950635023716, −6.91469103626909406223255043804, −6.44434197467298206476021389307, −6.15267013015603292112906972931, −5.25332741283121206700463308290, −4.10625798014363516984369255368, −3.80880328694342425730935973216, −2.80335944085781759882923856447, −2.44821729992400328434125228202, 0,
2.44821729992400328434125228202, 2.80335944085781759882923856447, 3.80880328694342425730935973216, 4.10625798014363516984369255368, 5.25332741283121206700463308290, 6.15267013015603292112906972931, 6.44434197467298206476021389307, 6.91469103626909406223255043804, 7.52219997040636286950635023716, 8.291996087541312922253124488251, 8.839886292016704738441339481672, 9.488699237661810053920246470193, 9.689914794558097820657325082483, 10.23151567133484212790849208076
