| L(s) = 1 | − 2·2-s + 2·4-s + 4·7-s + 6·9-s − 8·14-s − 4·16-s + 4·17-s − 12·18-s − 4·23-s + 10·25-s + 8·28-s − 16·31-s + 8·32-s − 8·34-s + 12·36-s − 12·41-s + 8·46-s − 4·47-s − 2·49-s − 20·50-s + 32·62-s + 24·63-s − 8·64-s + 8·68-s − 24·71-s − 20·73-s + 8·79-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.51·7-s + 2·9-s − 2.13·14-s − 16-s + 0.970·17-s − 2.82·18-s − 0.834·23-s + 2·25-s + 1.51·28-s − 2.87·31-s + 1.41·32-s − 1.37·34-s + 2·36-s − 1.87·41-s + 1.17·46-s − 0.583·47-s − 2/7·49-s − 2.82·50-s + 4.06·62-s + 3.02·63-s − 64-s + 0.970·68-s − 2.84·71-s − 2.34·73-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8059179076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8059179076\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−13.16814649393256031214318919516, −12.70518008338902403415164067872, −12.11796651426739510832669883137, −11.61906378473500750937600713502, −10.90008293744254375793420290761, −10.71175736328018571891666108158, −10.02855753845631006398792613196, −9.889490957767354348573297205681, −8.947174030589603036193396324195, −8.787308303653031155109193212186, −7.894991399041378037678351335258, −7.68554541075836599425581941410, −7.10535620975492105410554091608, −6.71783077344735857280363386585, −5.55335617011589335278110355886, −4.81775929522041497455260196203, −4.42086491609661664386152334463, −3.40324191097119578865247278690, −1.71995418139410474051515541141, −1.51481023897576554479171407969,
1.51481023897576554479171407969, 1.71995418139410474051515541141, 3.40324191097119578865247278690, 4.42086491609661664386152334463, 4.81775929522041497455260196203, 5.55335617011589335278110355886, 6.71783077344735857280363386585, 7.10535620975492105410554091608, 7.68554541075836599425581941410, 7.894991399041378037678351335258, 8.787308303653031155109193212186, 8.947174030589603036193396324195, 9.889490957767354348573297205681, 10.02855753845631006398792613196, 10.71175736328018571891666108158, 10.90008293744254375793420290761, 11.61906378473500750937600713502, 12.11796651426739510832669883137, 12.70518008338902403415164067872, 13.16814649393256031214318919516
