| L(s) = 1 | + 2·3-s + 2·7-s + 2·9-s + 2·13-s − 2·17-s + 8·19-s + 4·21-s − 10·23-s + 6·27-s + 10·37-s + 4·39-s + 12·41-s − 6·43-s − 14·47-s + 2·49-s − 4·51-s + 2·53-s + 16·57-s + 8·59-s + 4·61-s + 4·63-s + 14·67-s − 20·69-s − 18·73-s − 16·79-s + 11·81-s + 10·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s + 2/3·9-s + 0.554·13-s − 0.485·17-s + 1.83·19-s + 0.872·21-s − 2.08·23-s + 1.15·27-s + 1.64·37-s + 0.640·39-s + 1.87·41-s − 0.914·43-s − 2.04·47-s + 2/7·49-s − 0.560·51-s + 0.274·53-s + 2.11·57-s + 1.04·59-s + 0.512·61-s + 0.503·63-s + 1.71·67-s − 2.40·69-s − 2.10·73-s − 1.80·79-s + 11/9·81-s + 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.612513031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.612513031\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−10.18656293180407804554371281890, −10.06482760269775380371040344600, −9.578548788194541063102053656936, −9.253414012269336771168943351682, −8.525634426571618892253276351088, −8.493892987237767762568860655333, −7.930608750462495142989773829040, −7.68095093311166738795982055461, −7.21360138864090689770640421362, −6.64114012365709273266181391516, −5.98755994956523760813513073468, −5.78637342642598754756864453424, −4.92520628464722690442023488985, −4.64257544361796656521375668821, −3.94881765763319959652114720674, −3.58608957262901790098083189385, −2.90160514683644799411765705493, −2.39316962120136038256601805581, −1.73618760263876091414599865268, −0.956690192267222933516784088180,
0.956690192267222933516784088180, 1.73618760263876091414599865268, 2.39316962120136038256601805581, 2.90160514683644799411765705493, 3.58608957262901790098083189385, 3.94881765763319959652114720674, 4.64257544361796656521375668821, 4.92520628464722690442023488985, 5.78637342642598754756864453424, 5.98755994956523760813513073468, 6.64114012365709273266181391516, 7.21360138864090689770640421362, 7.68095093311166738795982055461, 7.930608750462495142989773829040, 8.493892987237767762568860655333, 8.525634426571618892253276351088, 9.253414012269336771168943351682, 9.578548788194541063102053656936, 10.06482760269775380371040344600, 10.18656293180407804554371281890
