L(s) = 1 | + (1.73 − i)3-s + (−1 + 1.73i)4-s − 4i·7-s + (0.499 − 0.866i)9-s + 3·11-s + 3.99i·12-s + (1.73 + i)13-s + (−1.99 − 3.46i)16-s + (5.19 − 3i)17-s + (3.5 − 2.59i)19-s + (−4 − 6.92i)21-s + 4.00i·27-s + (6.92 + 4i)28-s + (−1.5 + 2.59i)29-s − 7·31-s + ⋯ |
L(s) = 1 | + (0.999 − 0.577i)3-s + (−0.5 + 0.866i)4-s − 1.51i·7-s + (0.166 − 0.288i)9-s + 0.904·11-s + 1.15i·12-s + (0.480 + 0.277i)13-s + (−0.499 − 0.866i)16-s + (1.26 − 0.727i)17-s + (0.802 − 0.596i)19-s + (−0.872 − 1.51i)21-s + 0.769i·27-s + (1.30 + 0.755i)28-s + (−0.278 + 0.482i)29-s − 1.25·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.76655 - 0.509454i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.76655 - 0.509454i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-3.5 + 2.59i)T \) |
good | 2 | \( 1 + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.73 + i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + (-1.73 - i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-5.19 + 3i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.46 + 2i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.19 + 3i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.5 + 12.9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.73 + i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.5 - 2.59i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.92 - 4i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.92 + 4i)T + (48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.04334793401683353504668690531, −9.704515666828232234332986424584, −9.086983046505120545913033479251, −8.030173661203425597557555437492, −7.47027233913951022615622415120, −6.78498553458026951830260153804, −4.94418267041729288861672119069, −3.73746216017072337283333209722, −3.14190017737766234071291479024, −1.26052134014176786655065684227,
1.68963123435614751517504654443, 3.18598208237196360298998038068, 4.14581044886372423467508100544, 5.65087247543136885852418416393, 5.96004480878040017705807850892, 7.77464208771795972208962237532, 8.783436304268985795564881361405, 9.228234855902200390235690546355, 9.844501474706357922007565514408, 10.91381483453929153099001190199