| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s − 2·9-s − 2·14-s + 16-s − 6·17-s + 2·18-s + 25-s + 2·28-s − 8·31-s − 32-s + 6·34-s − 2·36-s − 6·41-s − 6·47-s + 3·49-s − 50-s − 2·56-s + 8·62-s − 4·63-s + 64-s − 6·68-s + 2·72-s + 4·73-s − 20·79-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.471·18-s + 1/5·25-s + 0.377·28-s − 1.43·31-s − 0.176·32-s + 1.02·34-s − 1/3·36-s − 0.937·41-s − 0.875·47-s + 3/7·49-s − 0.141·50-s − 0.267·56-s + 1.01·62-s − 0.503·63-s + 1/8·64-s − 0.727·68-s + 0.235·72-s + 0.468·73-s − 2.25·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.900967404214814360155728720045, −8.595391550004089428257614214162, −8.175114563020558380296865729629, −7.69476575482621482508469057464, −6.94526069192045777044790762582, −6.86171931732588786239234570025, −6.03311756257113652450391513724, −5.57607001516633241815407220328, −4.97470123691061072321941481938, −4.40871030244508749874393512611, −3.68296580336228982924419474549, −2.91521048978382309590202053643, −2.18735775250471208006222515460, −1.50991460483536013897516168917, 0,
1.50991460483536013897516168917, 2.18735775250471208006222515460, 2.91521048978382309590202053643, 3.68296580336228982924419474549, 4.40871030244508749874393512611, 4.97470123691061072321941481938, 5.57607001516633241815407220328, 6.03311756257113652450391513724, 6.86171931732588786239234570025, 6.94526069192045777044790762582, 7.69476575482621482508469057464, 8.175114563020558380296865729629, 8.595391550004089428257614214162, 8.900967404214814360155728720045
