L(s) = 1 | + (−1.39 + 0.211i)2-s + (1.91 − 0.591i)4-s + (−1.77 + 0.734i)5-s + (−0.0355 + 0.0355i)7-s + (−2.54 + 1.23i)8-s + (2.32 − 1.40i)10-s + (1.04 + 2.53i)11-s + (1.82 + 0.755i)13-s + (0.0422 − 0.0572i)14-s + (3.30 − 2.25i)16-s − 2.92i·17-s + (−1.55 − 0.642i)19-s + (−2.95 + 2.45i)20-s + (−2.00 − 3.31i)22-s + (−0.146 − 0.146i)23-s + ⋯ |
L(s) = 1 | + (−0.988 + 0.149i)2-s + (0.955 − 0.295i)4-s + (−0.792 + 0.328i)5-s + (−0.0134 + 0.0134i)7-s + (−0.900 + 0.435i)8-s + (0.734 − 0.443i)10-s + (0.315 + 0.762i)11-s + (0.506 + 0.209i)13-s + (0.0112 − 0.0153i)14-s + (0.825 − 0.564i)16-s − 0.710i·17-s + (−0.355 − 0.147i)19-s + (−0.660 + 0.547i)20-s + (−0.426 − 0.707i)22-s + (−0.0305 − 0.0305i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.129271 + 0.400799i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.129271 + 0.400799i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.39 - 0.211i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.77 - 0.734i)T + (3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (0.0355 - 0.0355i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.04 - 2.53i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-1.82 - 0.755i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 2.92iT - 17T^{2} \) |
| 19 | \( 1 + (1.55 + 0.642i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (0.146 + 0.146i)T + 23iT^{2} \) |
| 29 | \( 1 + (2.17 - 5.26i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 4.28T + 31T^{2} \) |
| 37 | \( 1 + (1.92 - 0.796i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (4.22 + 4.22i)T + 41iT^{2} \) |
| 43 | \( 1 + (-1.29 - 3.13i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 5.33iT - 47T^{2} \) |
| 53 | \( 1 + (1.48 + 3.59i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (1.27 - 0.529i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (5.58 - 13.4i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (-1.11 + 2.69i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (3.73 - 3.73i)T - 71iT^{2} \) |
| 73 | \( 1 + (10.8 + 10.8i)T + 73iT^{2} \) |
| 79 | \( 1 - 9.94iT - 79T^{2} \) |
| 83 | \( 1 + (-11.2 - 4.65i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (6.00 - 6.00i)T - 89iT^{2} \) |
| 97 | \( 1 + 3.69T + 97T^{2} \) |
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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
−10.54955525097467655898061260421, −9.462293734472982662018227869770, −8.935084102451156170887190547266, −7.890092519309685188334953273861, −7.23589332819804837332127821699, −6.56990792742411698572221944362, −5.39023854194354001645857559599, −4.07673390423156978915411971812, −2.92795378237300870845788020210, −1.54540079583163825361872990728,
0.28418431810258007443355478647, 1.76638457089515339213668425278, 3.30696593287043865588830537113, 4.09095405712853871443582123854, 5.69246647758034256197306689498, 6.50111650051610432421829365072, 7.56710030022095256530874683355, 8.315411852561255573237964974133, 8.772484335582861636979737509269, 9.804651017992622749123021800109