| L(s) = 1 | − 2·2-s + 3·4-s + 3·5-s − 4·8-s − 3·9-s − 6·10-s + 3·11-s − 13-s + 5·16-s + 3·17-s + 6·18-s − 7·19-s + 9·20-s − 6·22-s + 9·23-s + 5·25-s + 2·26-s − 3·29-s − 16·31-s − 6·32-s − 6·34-s − 9·36-s + 37-s + 14·38-s − 12·40-s + 3·41-s + 43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.34·5-s − 1.41·8-s − 9-s − 1.89·10-s + 0.904·11-s − 0.277·13-s + 5/4·16-s + 0.727·17-s + 1.41·18-s − 1.60·19-s + 2.01·20-s − 1.27·22-s + 1.87·23-s + 25-s + 0.392·26-s − 0.557·29-s − 2.87·31-s − 1.06·32-s − 1.02·34-s − 3/2·36-s + 0.164·37-s + 2.27·38-s − 1.89·40-s + 0.468·41-s + 0.152·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.084121015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.084121015\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.39562127406394720345447818060, −9.729209829164765977743811430845, −9.330575465001104408978983883949, −9.063372790816131529363498390117, −9.057205605416533279803614600937, −8.230039808944002442357392488973, −8.137300721426646937940238490548, −7.17869515237098545801502589486, −7.11108174311981492139832303749, −6.62053404230005211540171888006, −6.04688242461541621953646227894, −5.60617276254181397099908486755, −5.48515262991420957884108594787, −4.65000065925864479939546635242, −3.87426043143048719017810797897, −3.17609009421795780705131170886, −2.74438599493495838160807960792, −1.86662004822149684102865916908, −1.74079645369750714462172359529, −0.62175685291960500621511822822,
0.62175685291960500621511822822, 1.74079645369750714462172359529, 1.86662004822149684102865916908, 2.74438599493495838160807960792, 3.17609009421795780705131170886, 3.87426043143048719017810797897, 4.65000065925864479939546635242, 5.48515262991420957884108594787, 5.60617276254181397099908486755, 6.04688242461541621953646227894, 6.62053404230005211540171888006, 7.11108174311981492139832303749, 7.17869515237098545801502589486, 8.137300721426646937940238490548, 8.230039808944002442357392488973, 9.057205605416533279803614600937, 9.063372790816131529363498390117, 9.330575465001104408978983883949, 9.729209829164765977743811430845, 10.39562127406394720345447818060
