| L(s) = 1 | + 4·2-s + 3·3-s + 12·4-s − 4·5-s + 12·6-s + 32·8-s + 6·9-s − 16·10-s + 36·12-s + 7·13-s − 12·15-s + 80·16-s + 32·17-s + 24·18-s − 38·19-s − 48·20-s − 12·23-s + 96·24-s + 25·25-s + 28·26-s + 9·27-s − 52·29-s − 48·30-s + 192·32-s + 128·34-s + 72·36-s − 2·37-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3-s + 3·4-s − 4/5·5-s + 2·6-s + 4·8-s + 2/3·9-s − 8/5·10-s + 3·12-s + 7/13·13-s − 4/5·15-s + 5·16-s + 1.88·17-s + 4/3·18-s − 2·19-s − 2.39·20-s − 0.521·23-s + 4·24-s + 25-s + 1.07·26-s + 1/3·27-s − 1.79·29-s − 8/5·30-s + 6·32-s + 3.76·34-s + 2·36-s − 0.0540·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51984 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(10.18442945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.18442945\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T - 9 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 95 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 230 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 7 T - 120 T^{2} - 7 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 32 T + 735 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 12 T + 577 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 52 T + 1863 T^{2} + 52 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1775 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 44 T + 255 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 69 T + 3436 T^{2} + 69 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 90 T + 4909 T^{2} + 90 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 44 T - 873 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 54 T + 4453 T^{2} + 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T - 3600 T^{2} + 11 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 87 T + 7012 T^{2} + 87 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 78 T + 7069 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 5 T - 5304 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 129 T + 11788 T^{2} - 129 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 3550 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T + 8979 T^{2} + 130 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 74 T - 3933 T^{2} + 74 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−12.39828267809755689477669453106, −12.03631548331990587394543373768, −11.38147095946024126000815658196, −11.05970216035084154226412621657, −10.36743487688314476737462691473, −10.21384433743887461307843853001, −9.294537509451612653230297133136, −8.558058519681023139502935011581, −7.945804771910411765628817864380, −7.80459114747998402623657540609, −7.06315630583138195279521893690, −6.66491204497390342867495490748, −5.81377125225345865244036453895, −5.57181957223047742268519130492, −4.50726701651689037970905540335, −4.24972520740537973019194605743, −3.43891844253469826749389943943, −3.28893611731049360258257527438, −2.29187143021783766261748158087, −1.51232299841962825360412824158,
1.51232299841962825360412824158, 2.29187143021783766261748158087, 3.28893611731049360258257527438, 3.43891844253469826749389943943, 4.24972520740537973019194605743, 4.50726701651689037970905540335, 5.57181957223047742268519130492, 5.81377125225345865244036453895, 6.66491204497390342867495490748, 7.06315630583138195279521893690, 7.80459114747998402623657540609, 7.945804771910411765628817864380, 8.558058519681023139502935011581, 9.294537509451612653230297133136, 10.21384433743887461307843853001, 10.36743487688314476737462691473, 11.05970216035084154226412621657, 11.38147095946024126000815658196, 12.03631548331990587394543373768, 12.39828267809755689477669453106
