L(s) = 1 | + 16.2i·2-s + (−77.7 + 22.8i)3-s − 8.51·4-s + (−371. − 1.26e3i)6-s + 569.·7-s + 4.02e3i·8-s + (5.51e3 − 3.54e3i)9-s + 6.58e3i·11-s + (661. − 194. i)12-s + 2.82e4·13-s + 9.26e3i·14-s − 6.76e4·16-s + 1.51e5i·17-s + (5.77e4 + 8.97e4i)18-s + 4.15e4·19-s + ⋯ |
L(s) = 1 | + 1.01i·2-s + (−0.959 + 0.281i)3-s − 0.0332·4-s + (−0.286 − 0.975i)6-s + 0.237·7-s + 0.982i·8-s + (0.841 − 0.540i)9-s + 0.449i·11-s + (0.0319 − 0.00937i)12-s + 0.987·13-s + 0.241i·14-s − 1.03·16-s + 1.80i·17-s + (0.549 + 0.855i)18-s + 0.319·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.181868 - 1.26449i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.181868 - 1.26449i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (77.7 - 22.8i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - 16.2iT - 256T^{2} \) |
| 7 | \( 1 - 569.T + 5.76e6T^{2} \) |
| 11 | \( 1 - 6.58e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.82e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.51e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 4.15e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 1.73e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 7.83e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 2.15e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 2.71e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 6.76e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 4.21e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 8.87e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 6.53e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.54e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 8.68e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.87e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 3.73e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.75e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 4.22e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 7.28e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 5.99e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.15e7T + 7.83e15T^{2} \) |
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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
−13.60541963534216474132898285288, −12.27083039407793569494827368671, −11.21996101353373321939972241346, −10.30883506015621430527403354437, −8.687584365096294156160595568595, −7.43474402644993770822661579863, −6.26691500199819293640340121825, −5.49603937743414621138204650226, −4.02500188546850156437383345990, −1.58763424724869127216874403707,
0.46523941589574123321205434440, 1.54969399007558032006293292932, 3.16653320088241261532215280802, 4.80836687438759945431805688629, 6.26433325658989462211366491774, 7.40254606084332952162241801562, 9.199274917543119056954515909319, 10.54851965045157321879733085543, 11.22132424056109384246882934472, 12.00888463313603645828785686031