| L(s) = 1 | + 4·5-s + 2·7-s + 3·9-s + 2·11-s − 6·17-s + 6·19-s − 8·23-s + 2·25-s − 2·29-s + 20·31-s + 8·35-s + 6·37-s + 6·41-s − 4·43-s + 12·45-s − 4·47-s + 7·49-s + 12·53-s + 8·55-s + 10·59-s + 2·61-s + 6·63-s − 10·67-s − 10·71-s + 4·73-s + 4·77-s − 8·79-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.755·7-s + 9-s + 0.603·11-s − 1.45·17-s + 1.37·19-s − 1.66·23-s + 2/5·25-s − 0.371·29-s + 3.59·31-s + 1.35·35-s + 0.986·37-s + 0.937·41-s − 0.609·43-s + 1.78·45-s − 0.583·47-s + 49-s + 1.64·53-s + 1.07·55-s + 1.30·59-s + 0.256·61-s + 0.755·63-s − 1.22·67-s − 1.18·71-s + 0.468·73-s + 0.455·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.519221445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.519221445\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 10 T + 29 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.39692297982278897478112204879, −10.15896970138293606442782531757, −9.790276531382174952774325682804, −9.752585407110636031026243756694, −8.943451830316586849713500732680, −8.777725825795076814792215086605, −7.967323186872632161068494912375, −7.82177861848072737912835206974, −7.10795890590299412586677499086, −6.58314087253901998840300588561, −6.28652908579985239071975431083, −5.80906191205362523421399047185, −5.38386760921177306659848086547, −4.73955054617102410807352979030, −4.11932333262077994974457587944, −4.05815452372245929095434204690, −2.60947219795475961566357644966, −2.48011681946557116919175285496, −1.60700338308445707531075240958, −1.17455097543999622918973221100,
1.17455097543999622918973221100, 1.60700338308445707531075240958, 2.48011681946557116919175285496, 2.60947219795475961566357644966, 4.05815452372245929095434204690, 4.11932333262077994974457587944, 4.73955054617102410807352979030, 5.38386760921177306659848086547, 5.80906191205362523421399047185, 6.28652908579985239071975431083, 6.58314087253901998840300588561, 7.10795890590299412586677499086, 7.82177861848072737912835206974, 7.967323186872632161068494912375, 8.777725825795076814792215086605, 8.943451830316586849713500732680, 9.752585407110636031026243756694, 9.790276531382174952774325682804, 10.15896970138293606442782531757, 10.39692297982278897478112204879
