| L(s) = 1 | + 18.4·2-s + (33.7 + 58.5i)3-s + 212.·4-s + (54.0 + 93.5i)5-s + (623. + 1.08e3i)6-s + (395. − 685. i)7-s + 1.56e3·8-s + (−1.19e3 + 2.06e3i)9-s + (996. + 1.72e3i)10-s − 1.37e3·11-s + (7.19e3 + 1.24e4i)12-s + (1.30e3 − 2.26e3i)13-s + (7.30e3 − 1.26e4i)14-s + (−3.64e3 + 6.32e3i)15-s + 1.68e3·16-s + (−1.03e4 + 1.79e4i)17-s + ⋯ |
| L(s) = 1 | + 1.63·2-s + (0.722 + 1.25i)3-s + 1.66·4-s + (0.193 + 0.334i)5-s + (1.17 + 2.04i)6-s + (0.436 − 0.755i)7-s + 1.08·8-s + (−0.544 + 0.942i)9-s + (0.315 + 0.546i)10-s − 0.310·11-s + (1.20 + 2.08i)12-s + (0.164 − 0.285i)13-s + (0.711 − 1.23i)14-s + (−0.279 + 0.483i)15-s + 0.102·16-s + (−0.510 + 0.884i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.85554 + 2.68324i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.85554 + 2.68324i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-5.02e5 + 1.37e5i)T \) |
| good | 2 | \( 1 - 18.4T + 128T^{2} \) |
| 3 | \( 1 + (-33.7 - 58.5i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-54.0 - 93.5i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-395. + 685. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + 1.37e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-1.30e3 + 2.26e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (1.03e4 - 1.79e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.74e3 + 4.75e3i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (3.12e4 + 5.41e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-7.86e4 + 1.36e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-1.06e5 - 1.83e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (4.94e4 + 8.57e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 - 1.10e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 1.22e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + (6.08e5 + 1.05e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + 1.40e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + (-1.57e5 + 2.73e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-9.60e5 - 1.66e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.15e6 + 2.00e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (3.17e6 - 5.49e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (1.26e6 - 2.19e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-2.92e6 - 5.07e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-3.84e6 - 6.66e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + 1.14e7T + 8.07e13T^{2} \) |
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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
−14.44571672940641272604753696764, −13.90313957247665336962242959696, −12.62334880136841887865275784530, −10.98884610386923979074200860008, −10.17618422709120154837067731670, −8.355318202815710627407853008965, −6.44619396103093075495787026529, −4.76805962508984659401220987063, −3.94608913684938049734525595391, −2.63913649895902345556129332572,
1.76944622652194910492739063055, 2.94388295952144215829077612201, 4.89645149673136153172311385959, 6.23398759210172696771653884781, 7.56362323546555117034006232078, 8.985750166395862428035701698267, 11.43906788759994828861062921971, 12.35922212268728605616135304445, 13.25700220015644408137671128538, 13.93120574300172811307430114251
