L(s) = 1 | − 2.15i·2-s + 11.3·4-s − 30.9·5-s + 73.3·7-s − 58.9i·8-s + 66.6i·10-s + 203. i·11-s − 115. i·13-s − 158. i·14-s + 54.5·16-s + 149.·17-s − 97.5·19-s − 351.·20-s + 438.·22-s + 785. i·23-s + ⋯ |
L(s) = 1 | − 0.538i·2-s + 0.709·4-s − 1.23·5-s + 1.49·7-s − 0.921i·8-s + 0.666i·10-s + 1.68i·11-s − 0.681i·13-s − 0.806i·14-s + 0.213·16-s + 0.517·17-s − 0.270·19-s − 0.878·20-s + 0.906·22-s + 1.48i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.295033880\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.295033880\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 + (3.34e3 - 964. i)T \) |
good | 2 | \( 1 + 2.15iT - 16T^{2} \) |
| 5 | \( 1 + 30.9T + 625T^{2} \) |
| 7 | \( 1 - 73.3T + 2.40e3T^{2} \) |
| 11 | \( 1 - 203. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 115. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 149.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 97.5T + 1.30e5T^{2} \) |
| 23 | \( 1 - 785. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 279.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 520. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 737. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 540.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.17e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.50e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.63e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 5.34e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 1.69e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 9.06e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 876. iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 7.89e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 3.53e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.14e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.51e4iT - 8.85e7T^{2} \) |
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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
−10.53405884096909846596336248247, −9.632153722560381914986846224720, −8.197274039085538593410218379357, −7.60556479082490234606526769999, −7.03487796935482620504076913517, −5.40323946511880730757742189472, −4.42383930254365471146793722652, −3.48013571930391607540563206154, −2.10462444601589398919402014083, −1.16139616474366652450633116961,
0.63984056594571631474794570034, 2.09762622900116108416973577820, 3.48522627305143533325073655237, 4.57934265155524409952720563847, 5.63752182008282626456701694398, 6.63769523946134443888880509401, 7.71811719758424524079715150158, 8.156687393717238292008210413151, 8.816898037918575860665467876897, 10.65149886439106021082418792626