| L(s) = 1 | + (−10.6 + 6.14i)5-s + (7.14 − 12.3i)7-s + (−90.2 − 52.0i)11-s + (−37.6 − 65.1i)13-s − 341. i·17-s − 706.·19-s + (516. − 298. i)23-s + (−237. + 410. i)25-s + (−1.12e3 − 651. i)29-s + (−514. − 891. i)31-s + 175. i·35-s + 563.·37-s + (−85.8 + 49.5i)41-s + (448. − 776. i)43-s + (−372. − 215. i)47-s + ⋯ |
| L(s) = 1 | + (−0.425 + 0.245i)5-s + (0.145 − 0.252i)7-s + (−0.745 − 0.430i)11-s + (−0.222 − 0.385i)13-s − 1.18i·17-s − 1.95·19-s + (0.976 − 0.563i)23-s + (−0.379 + 0.657i)25-s + (−1.34 − 0.774i)29-s + (−0.535 − 0.927i)31-s + 0.143i·35-s + 0.411·37-s + (−0.0510 + 0.0294i)41-s + (0.242 − 0.419i)43-s + (−0.168 − 0.0974i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.211921 - 0.559470i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.211921 - 0.559470i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (10.6 - 6.14i)T + (312.5 - 541. i)T^{2} \) |
| 7 | \( 1 + (-7.14 + 12.3i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (90.2 + 52.0i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (37.6 + 65.1i)T + (-1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + 341. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 706.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-516. + 298. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (1.12e3 + 651. i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (514. + 891. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 563.T + 1.87e6T^{2} \) |
| 41 | \( 1 + (85.8 - 49.5i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-448. + 776. i)T + (-1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (372. + 215. i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 - 5.27e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (-4.88e3 + 2.81e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (565. - 979. i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-676. - 1.17e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 5.68e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.23e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-3.06e3 + 5.31e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (6.50e3 + 3.75e3i)T + (2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 - 8.72e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (2.72e3 - 4.71e3i)T + (-4.42e7 - 7.66e7i)T^{2} \) |
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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
−12.71488456433569075815578670651, −11.32655243337715812596621894530, −10.69036188578142727889393103710, −9.330623820302314809490041364441, −8.065043980709796262134856907268, −7.10884661743073018683087336789, −5.61060217594513426481135999985, −4.17445744714903414580955188443, −2.57642796736828470645343009869, −0.25181078929944249186948372453,
2.02686155251008694596792019030, 3.93077098241084770941989653096, 5.23363382367211402946438630424, 6.71060114834652450625929082702, 8.013861801341204167727120487863, 8.931108773078019095731227093512, 10.33081129273720379339018965693, 11.24383880767117004219228915309, 12.54926106192211155431958350472, 13.06884891300781696216993948461
