L(s) = 1 | + 15·4-s − 9·9-s + 22·11-s + 161·16-s + 120·19-s + 180·29-s − 16·31-s − 135·36-s + 844·41-s + 330·44-s + 10·49-s + 400·59-s + 264·61-s + 1.45e3·64-s + 1.52e3·71-s + 1.80e3·76-s + 1.10e3·79-s + 81·81-s − 1.14e3·89-s − 198·99-s + 3.40e3·101-s + 640·109-s + 2.70e3·116-s + 363·121-s − 240·124-s + 127-s + 131-s + ⋯ |
L(s) = 1 | + 15/8·4-s − 1/3·9-s + 0.603·11-s + 2.51·16-s + 1.44·19-s + 1.15·29-s − 0.0926·31-s − 5/8·36-s + 3.21·41-s + 1.13·44-s + 0.0291·49-s + 0.882·59-s + 0.554·61-s + 2.84·64-s + 2.54·71-s + 2.71·76-s + 1.56·79-s + 1/9·81-s − 1.35·89-s − 0.201·99-s + 3.35·101-s + 0.562·109-s + 2.16·116-s + 3/11·121-s − 0.173·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(7.794247071\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.794247071\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
good | 2 | $C_2^2$ | \( 1 - 15 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3370 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4350 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 60 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 8790 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 90 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 96950 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 422 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 7450 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 48390 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 176650 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 200 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 132 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 471770 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 762 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 484270 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 550 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1126150 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 570 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1825150 T^{2} + p^{6} T^{4} \) |
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
−10.01120333756950394034449735719, −9.796455685942490626700695552492, −9.198672754388914411758559260146, −8.860100217433436023152778046512, −8.079264862538169184606742133672, −7.918501084000657947375197863043, −7.31119076985933211315344881999, −7.18190199638508235648026802132, −6.41323530984700875373023987972, −6.33485141451524277385433051572, −5.80459015168028023573784081393, −5.28928287962058318132939894842, −4.81827071383428850865880771738, −3.88735800724838438680904365490, −3.60151395813606525078454654794, −2.87264938533099546527297128886, −2.50121057571767823018062024288, −2.01719518082683411702678884092, −1.01986425254504192699302747427, −0.900957363883468340896770641342,
0.900957363883468340896770641342, 1.01986425254504192699302747427, 2.01719518082683411702678884092, 2.50121057571767823018062024288, 2.87264938533099546527297128886, 3.60151395813606525078454654794, 3.88735800724838438680904365490, 4.81827071383428850865880771738, 5.28928287962058318132939894842, 5.80459015168028023573784081393, 6.33485141451524277385433051572, 6.41323530984700875373023987972, 7.18190199638508235648026802132, 7.31119076985933211315344881999, 7.918501084000657947375197863043, 8.079264862538169184606742133672, 8.860100217433436023152778046512, 9.198672754388914411758559260146, 9.796455685942490626700695552492, 10.01120333756950394034449735719