| L(s) = 1 | − 2·2-s + 2·4-s + 2·5-s + 9-s − 4·10-s − 12·13-s − 4·16-s + 6·17-s − 2·18-s + 4·20-s − 7·25-s + 24·26-s − 20·29-s + 8·32-s − 12·34-s + 2·36-s + 16·37-s − 16·41-s + 2·45-s − 5·49-s + 14·50-s − 24·52-s − 12·53-s + 40·58-s + 14·61-s − 8·64-s − 24·65-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.894·5-s + 1/3·9-s − 1.26·10-s − 3.32·13-s − 16-s + 1.45·17-s − 0.471·18-s + 0.894·20-s − 7/5·25-s + 4.70·26-s − 3.71·29-s + 1.41·32-s − 2.05·34-s + 1/3·36-s + 2.63·37-s − 2.49·41-s + 0.298·45-s − 5/7·49-s + 1.97·50-s − 3.32·52-s − 1.64·53-s + 5.25·58-s + 1.79·61-s − 64-s − 2.97·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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
−9.721879645093847090940062433997, −9.400588261603299107064903686230, −9.256845722173249154900109293870, −8.062798749262723709584757589645, −7.75525037135579232003734806978, −7.42319981131404835915630105083, −7.04173846168624158743830695656, −6.11989947342759976206118717321, −5.49960710074578838011452836914, −5.02993077790447449830764321556, −4.29287803140015808576580756413, −3.22481329079945918066760847835, −2.13307257724406321246824925475, −1.85542976329431191785537331479, 0,
1.85542976329431191785537331479, 2.13307257724406321246824925475, 3.22481329079945918066760847835, 4.29287803140015808576580756413, 5.02993077790447449830764321556, 5.49960710074578838011452836914, 6.11989947342759976206118717321, 7.04173846168624158743830695656, 7.42319981131404835915630105083, 7.75525037135579232003734806978, 8.062798749262723709584757589645, 9.256845722173249154900109293870, 9.400588261603299107064903686230, 9.721879645093847090940062433997
