| L(s) = 1 | + 2·2-s + 2·4-s + 2·5-s + 4·7-s − 5·9-s + 4·10-s − 11-s + 8·14-s − 4·16-s − 10·18-s + 4·20-s − 2·22-s − 7·25-s + 8·28-s − 8·32-s + 8·35-s − 10·36-s + 6·37-s + 12·43-s − 2·44-s − 10·45-s − 2·49-s − 14·50-s − 12·53-s − 2·55-s − 20·63-s − 8·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.894·5-s + 1.51·7-s − 5/3·9-s + 1.26·10-s − 0.301·11-s + 2.13·14-s − 16-s − 2.35·18-s + 0.894·20-s − 0.426·22-s − 7/5·25-s + 1.51·28-s − 1.41·32-s + 1.35·35-s − 5/3·36-s + 0.986·37-s + 1.82·43-s − 0.301·44-s − 1.49·45-s − 2/7·49-s − 1.97·50-s − 1.64·53-s − 0.269·55-s − 2.51·63-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.566640553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.566640553\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−10.88023677523055899886301116177, −10.76517194615392112496312976161, −9.531323384279074079649336150802, −9.366222803316740054530803267641, −8.595888941067365837350463793393, −7.81893826163275369649161903506, −7.78789607043645765017922373874, −6.29323672924194667424631097798, −6.23870064789214498217427813253, −5.41175071356617719099292784896, −5.16796004190121525070885092557, −4.44574490854809981150981932393, −3.57632858228471856718550416679, −2.63044898935838963010520851422, −2.01168702564605014700712887669,
2.01168702564605014700712887669, 2.63044898935838963010520851422, 3.57632858228471856718550416679, 4.44574490854809981150981932393, 5.16796004190121525070885092557, 5.41175071356617719099292784896, 6.23870064789214498217427813253, 6.29323672924194667424631097798, 7.78789607043645765017922373874, 7.81893826163275369649161903506, 8.595888941067365837350463793393, 9.366222803316740054530803267641, 9.531323384279074079649336150802, 10.76517194615392112496312976161, 10.88023677523055899886301116177
