L(s) = 1 | + (−3.54 − 6.13i)2-s + (−9.12 + 15.8i)4-s + (−41.1 − 71.1i)5-s + (112. − 64.3i)7-s − 97.4·8-s + (−291. + 504. i)10-s + (176. − 305. i)11-s − 885.·13-s + (−793. − 463. i)14-s + (637. + 1.10e3i)16-s + (−212. + 368. i)17-s + (781. + 1.35e3i)19-s + 1.50e3·20-s − 2.49e3·22-s + (1.39e3 + 2.41e3i)23-s + ⋯ |
L(s) = 1 | + (−0.626 − 1.08i)2-s + (−0.285 + 0.494i)4-s + (−0.735 − 1.27i)5-s + (0.868 − 0.496i)7-s − 0.538·8-s + (−0.921 + 1.59i)10-s + (0.438 − 0.760i)11-s − 1.45·13-s + (−1.08 − 0.631i)14-s + (0.622 + 1.07i)16-s + (−0.178 + 0.308i)17-s + (0.496 + 0.859i)19-s + 0.839·20-s − 1.09·22-s + (0.549 + 0.951i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.320585 + 0.514224i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.320585 + 0.514224i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-112. + 64.3i)T \) |
good | 2 | \( 1 + (3.54 + 6.13i)T + (-16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (41.1 + 71.1i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-176. + 305. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + 885.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (212. - 368. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-781. - 1.35e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-1.39e3 - 2.41e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 3.67e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-1.79e3 + 3.11e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (7.14e3 + 1.23e4i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.43e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.58e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.88e3 + 4.99e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.26e4 - 2.19e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-2.16e4 + 3.74e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (9.72e3 + 1.68e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.47e4 + 2.54e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 5.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (1.89e4 - 3.28e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (2.68e4 + 4.65e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 8.59e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (1.04e4 + 1.80e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 9.75e4T + 8.58e9T^{2} \) |
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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
−12.70847034076101533692204859666, −11.86805905669546996842840234828, −11.10768884600883979281786372526, −9.732824789717839361409993371092, −8.694732183879807184656204730775, −7.65076305140642413957601628306, −5.27976850435590782613626765845, −3.78453504970872479311715876993, −1.61384001157345353117703984267, −0.35040597774110444556681253239,
2.75374132522611519442951574506, 4.94242749797742854067167367827, 6.85104592006000596569391144458, 7.32880044227816268214429183640, 8.568477263583231068181716581836, 9.903986808683301795019800814959, 11.39420469796954768048672122803, 12.19056192415167850125472106804, 14.33319416665111161822256248570, 14.99588792687239974060142834035