| 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
−16.37798028972311270817207601579, −15.39838545800286061620551721972, −12.82152692878901315608294919232, −11.59356743341455053436087511161, −10.22414829544985341380687092607, −9.319026777559085958297394248137, −8.305101650520161030511797785269, −5.32539932262742369394584451183, −2.60005455964037568351486951287, −0.28595968277332979834181896463,
1.69753762143710169504592749484, 6.32690821001983921277784695141, 7.02052528806474892105472095740, 8.391362574289126604448769619411, 10.15172495215270272239098639911, 11.16807973118008463044189723636, 13.51331798146089750324820387738, 14.63030003913393118872991238715, 16.40545679227864789327305459680, 17.24138593156365052475302899232
