L(s) = 1 | + (−1.5 + 0.866i)3-s + (−3.25 − 5.64i)5-s + 11.0·7-s + (1.5 − 2.59i)9-s + 17.2·11-s + (13.2 + 7.66i)13-s + (9.77 + 5.64i)15-s + (−12.0 − 20.9i)17-s + (−16.6 + 9.07i)19-s + (−16.5 + 9.53i)21-s + (−12.6 + 21.9i)23-s + (−8.74 + 15.1i)25-s + 5.19i·27-s + (10.6 + 6.17i)29-s − 15.5i·31-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.288i)3-s + (−0.651 − 1.12i)5-s + 1.57·7-s + (0.166 − 0.288i)9-s + 1.56·11-s + (1.02 + 0.589i)13-s + (0.651 + 0.376i)15-s + (−0.711 − 1.23i)17-s + (−0.878 + 0.477i)19-s + (−0.786 + 0.454i)21-s + (−0.551 + 0.955i)23-s + (−0.349 + 0.605i)25-s + 0.192i·27-s + (0.368 + 0.212i)29-s − 0.502i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.602i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.797 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.885297471\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.885297471\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 19 | \( 1 + (16.6 - 9.07i)T \) |
good | 5 | \( 1 + (3.25 + 5.64i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 - 11.0T + 49T^{2} \) |
| 11 | \( 1 - 17.2T + 121T^{2} \) |
| 13 | \( 1 + (-13.2 - 7.66i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (12.0 + 20.9i)T + (-144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (12.6 - 21.9i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-10.6 - 6.17i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + 15.5iT - 961T^{2} \) |
| 37 | \( 1 - 17.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-43.6 + 25.1i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-31.6 - 54.8i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-35.1 + 60.8i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (12.8 + 7.40i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-59.8 + 34.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-18.6 + 32.2i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-85.3 - 49.2i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (37.5 - 21.6i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (48.2 + 83.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (40.1 - 23.1i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 66.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-32.5 - 18.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (5.11 - 2.95i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.634129932053223898439564054379, −8.809318556744829272890622531784, −8.392948022520674819263754649315, −7.32670053440175452604322531933, −6.27042303926118558780028374290, −5.24007909581122124246365222802, −4.28596175129995189106771268287, −4.05099472073517771901187802616, −1.77081597085415917246229203745, −0.860333751985016733787490310128,
1.10549134334923535469236117714, 2.29356575555278049449524736632, 3.91076098287910909587854658723, 4.39017206417581032211284968567, 5.91213927507058384969828999250, 6.52126406203268297988304598150, 7.37614957668691843060146763606, 8.315411206934206288955955116311, 8.820962903783325696499421292889, 10.49764868604211439693941497907