| L(s) = 1 | + (−17.0 + 9.84i)2-s + (−12.0 − 45.1i)3-s + (130. − 225. i)4-s + (147. + 255. i)5-s + (651. + 651. i)6-s + (−45.1 − 906. i)7-s + 2.60e3i·8-s + (−1.89e3 + 1.09e3i)9-s + (−5.03e3 − 2.90e3i)10-s + (−2.53e3 − 1.46e3i)11-s + (−1.17e4 − 3.15e3i)12-s + 1.24e4i·13-s + (9.69e3 + 1.50e4i)14-s + (9.76e3 − 9.75e3i)15-s + (−8.97e3 − 1.55e4i)16-s + (−1.37e4 + 2.38e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.50 + 0.870i)2-s + (−0.258 − 0.966i)3-s + (1.01 − 1.75i)4-s + (0.528 + 0.914i)5-s + (1.23 + 1.23i)6-s + (−0.0497 − 0.998i)7-s + 1.79i·8-s + (−0.866 + 0.499i)9-s + (−1.59 − 0.919i)10-s + (−0.573 − 0.330i)11-s + (−1.96 − 0.526i)12-s + 1.57i·13-s + (0.944 + 1.46i)14-s + (0.746 − 0.746i)15-s + (−0.547 − 0.949i)16-s + (−0.678 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 - 0.572i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.820 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0948180 + 0.301615i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0948180 + 0.301615i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (12.0 + 45.1i)T \) |
| 7 | \( 1 + (45.1 + 906. i)T \) |
| good | 2 | \( 1 + (17.0 - 9.84i)T + (64 - 110. i)T^{2} \) |
| 5 | \( 1 + (-147. - 255. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (2.53e3 + 1.46e3i)T + (9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 - 1.24e4iT - 6.27e7T^{2} \) |
| 17 | \( 1 + (1.37e4 - 2.38e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (3.84e3 - 2.21e3i)T + (4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (3.40e4 - 1.96e4i)T + (1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 - 1.04e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 + (3.49e3 + 2.02e3i)T + (1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-1.88e5 - 3.26e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + 2.34e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.55e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (4.50e4 + 7.80e4i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-4.01e5 - 2.31e5i)T + (5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.12e5 + 1.94e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.12e6 + 1.22e6i)T + (1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (2.21e6 - 3.82e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + 4.85e6iT - 9.09e12T^{2} \) |
| 73 | \( 1 + (-1.42e6 - 8.20e5i)T + (5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (5.86e5 + 1.01e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + 7.03e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (2.11e5 + 3.67e5i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + 4.84e6iT - 8.07e13T^{2} \) |
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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
−17.24138593156365052475302899232, −16.40545679227864789327305459680, −14.63030003913393118872991238715, −13.51331798146089750324820387738, −11.16807973118008463044189723636, −10.15172495215270272239098639911, −8.391362574289126604448769619411, −7.02052528806474892105472095740, −6.32690821001983921277784695141, −1.69753762143710169504592749484,
0.28595968277332979834181896463, 2.60005455964037568351486951287, 5.32539932262742369394584451183, 8.305101650520161030511797785269, 9.319026777559085958297394248137, 10.22414829544985341380687092607, 11.59356743341455053436087511161, 12.82152692878901315608294919232, 15.39838545800286061620551721972, 16.37798028972311270817207601579
