L(s) = 1 | − i·2-s + 3i·3-s + 7·4-s + 3·6-s − 26i·7-s − 15i·8-s − 9·9-s + 11·11-s + 21i·12-s + 32i·13-s − 26·14-s + 41·16-s + 74i·17-s + 9i·18-s + 60·19-s + ⋯ |
L(s) = 1 | − 0.353i·2-s + 0.577i·3-s + 0.875·4-s + 0.204·6-s − 1.40i·7-s − 0.662i·8-s − 0.333·9-s + 0.301·11-s + 0.505i·12-s + 0.682i·13-s − 0.496·14-s + 0.640·16-s + 1.05i·17-s + 0.117i·18-s + 0.724·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.791817879\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.791817879\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
good | 2 | \( 1 + iT - 8T^{2} \) |
| 7 | \( 1 + 26iT - 343T^{2} \) |
| 13 | \( 1 - 32iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 74iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 60T + 6.85e3T^{2} \) |
| 23 | \( 1 - 182iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 90T + 2.43e4T^{2} \) |
| 31 | \( 1 + 8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 66iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 422T + 6.89e4T^{2} \) |
| 43 | \( 1 + 408iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 506iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 348iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 200T + 2.05e5T^{2} \) |
| 61 | \( 1 - 132T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.03e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 762T + 3.57e5T^{2} \) |
| 73 | \( 1 - 542iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 550T + 4.93e5T^{2} \) |
| 83 | \( 1 - 132iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 570T + 7.04e5T^{2} \) |
| 97 | \( 1 - 14iT - 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.01120333756950394034449735719, −9.198672754388914411758559260146, −7.918501084000657947375197863043, −7.18190199638508235648026802132, −6.41323530984700875373023987972, −5.28928287962058318132939894842, −3.88735800724838438680904365490, −3.60151395813606525078454654794, −2.01719518082683411702678884092, −0.900957363883468340896770641342,
1.01986425254504192699302747427, 2.50121057571767823018062024288, 2.87264938533099546527297128886, 4.81827071383428850865880771738, 5.80459015168028023573784081393, 6.33485141451524277385433051572, 7.31119076985933211315344881999, 8.079264862538169184606742133672, 8.860100217433436023152778046512, 9.796455685942490626700695552492