| L(s) = 1 | + (4.37 − 5.48i)2-s + (−9.64 − 12.0i)3-s + (−3.82 − 16.7i)4-s + (−28.9 − 13.9i)5-s − 108.·6-s − 153.·7-s + (93.5 + 45.0i)8-s + (0.847 − 3.71i)9-s + (−202. + 97.6i)10-s + (−14.7 + 64.7i)11-s + (−165. + 207. i)12-s + (−483. − 232. i)13-s + (−671. + 842. i)14-s + (110. + 483. i)15-s + (1.15e3 − 554. i)16-s + (65.9 − 31.7i)17-s + ⋯ |
| L(s) = 1 | + (0.773 − 0.969i)2-s + (−0.618 − 0.775i)3-s + (−0.119 − 0.524i)4-s + (−0.517 − 0.249i)5-s − 1.23·6-s − 1.18·7-s + (0.516 + 0.248i)8-s + (0.00348 − 0.0152i)9-s + (−0.641 + 0.308i)10-s + (−0.0368 + 0.161i)11-s + (−0.332 + 0.416i)12-s + (−0.793 − 0.381i)13-s + (−0.916 + 1.14i)14-s + (0.126 + 0.555i)15-s + (1.12 − 0.541i)16-s + (0.0553 − 0.0266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.200725 + 1.11665i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.200725 + 1.11665i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.05e4 - 5.88e3i)T \) |
| good | 2 | \( 1 + (-4.37 + 5.48i)T + (-7.12 - 31.1i)T^{2} \) |
| 3 | \( 1 + (9.64 + 12.0i)T + (-54.0 + 236. i)T^{2} \) |
| 5 | \( 1 + (28.9 + 13.9i)T + (1.94e3 + 2.44e3i)T^{2} \) |
| 7 | \( 1 + 153.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (14.7 - 64.7i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (483. + 232. i)T + (2.31e5 + 2.90e5i)T^{2} \) |
| 17 | \( 1 + (-65.9 + 31.7i)T + (8.85e5 - 1.11e6i)T^{2} \) |
| 19 | \( 1 + (275. + 1.20e3i)T + (-2.23e6 + 1.07e6i)T^{2} \) |
| 23 | \( 1 + (-392. + 1.71e3i)T + (-5.79e6 - 2.79e6i)T^{2} \) |
| 29 | \( 1 + (-3.84e3 + 4.81e3i)T + (-4.56e6 - 1.99e7i)T^{2} \) |
| 31 | \( 1 + (-1.02e3 + 1.28e3i)T + (-6.37e6 - 2.79e7i)T^{2} \) |
| 37 | \( 1 - 6.03e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-2.24e3 + 2.80e3i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 47 | \( 1 + (-3.93e3 - 1.72e4i)T + (-2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (1.46e4 - 7.04e3i)T + (2.60e8 - 3.26e8i)T^{2} \) |
| 59 | \( 1 + (2.87e4 - 1.38e4i)T + (4.45e8 - 5.58e8i)T^{2} \) |
| 61 | \( 1 + (9.24e3 + 1.15e4i)T + (-1.87e8 + 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-1.72e3 - 7.56e3i)T + (-1.21e9 + 5.85e8i)T^{2} \) |
| 71 | \( 1 + (9.58e3 + 4.19e4i)T + (-1.62e9 + 7.82e8i)T^{2} \) |
| 73 | \( 1 + (-3.23e4 - 1.55e4i)T + (1.29e9 + 1.62e9i)T^{2} \) |
| 79 | \( 1 - 5.57e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-7.27e4 - 9.11e4i)T + (-8.76e8 + 3.84e9i)T^{2} \) |
| 89 | \( 1 + (5.42e4 + 6.80e4i)T + (-1.24e9 + 5.44e9i)T^{2} \) |
| 97 | \( 1 + (-1.97e4 + 8.66e4i)T + (-7.73e9 - 3.72e9i)T^{2} \) |
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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
−13.73911425178176116796833360509, −12.53614517231614244331793578469, −12.40441680681076586113754092951, −11.14673906361359846109676400668, −9.726877055671674125834047727525, −7.65681335178096637575465208513, −6.24229009588544501053871045459, −4.42058170508736327725192440291, −2.74367852239952547963267994158, −0.51115869709720880703035943847,
3.72336163068330086371427357945, 5.07830796599025050583204796942, 6.32289221106338084841989476857, 7.55328065828153846825901402594, 9.678904210002677449180041594061, 10.72762331738618890932364089871, 12.23768762350503568956458663897, 13.49220543096936361464073714308, 14.72086439112074073513133548778, 15.73530563369987703042249466819
