| L(s) = 1 | + (−2.91 + 6.04i)2-s + (50.5 + 3.78i)3-s + (11.7 + 14.7i)4-s + (−30.3 + 98.4i)5-s + (−170. + 294. i)6-s + (−190. + 109. i)7-s + (−542. + 123. i)8-s + (1.82e3 + 274. i)9-s + (−507. − 470. i)10-s + (1.13e3 − 1.41e3i)11-s + (540. + 792. i)12-s + (−1.20e3 + 1.11e3i)13-s + (−110. − 1.47e3i)14-s + (−1.90e3 + 4.86e3i)15-s + (562. − 2.46e3i)16-s + (−212. + 65.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.364 + 0.756i)2-s + (1.87 + 0.140i)3-s + (0.184 + 0.231i)4-s + (−0.242 + 0.787i)5-s + (−0.788 + 1.36i)6-s + (−0.555 + 0.320i)7-s + (−1.06 + 0.241i)8-s + (2.49 + 0.376i)9-s + (−0.507 − 0.470i)10-s + (0.849 − 1.06i)11-s + (0.312 + 0.458i)12-s + (−0.548 + 0.508i)13-s + (−0.0402 − 0.536i)14-s + (−0.565 + 1.44i)15-s + (0.137 − 0.601i)16-s + (−0.0432 + 0.0133i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.34911 + 2.15805i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34911 + 2.15805i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-5.97e4 + 5.24e4i)T \) |
| good | 2 | \( 1 + (2.91 - 6.04i)T + (-39.9 - 50.0i)T^{2} \) |
| 3 | \( 1 + (-50.5 - 3.78i)T + (720. + 108. i)T^{2} \) |
| 5 | \( 1 + (30.3 - 98.4i)T + (-1.29e4 - 8.80e3i)T^{2} \) |
| 7 | \( 1 + (190. - 109. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.13e3 + 1.41e3i)T + (-3.94e5 - 1.72e6i)T^{2} \) |
| 13 | \( 1 + (1.20e3 - 1.11e3i)T + (3.60e5 - 4.81e6i)T^{2} \) |
| 17 | \( 1 + (212. - 65.5i)T + (1.99e7 - 1.35e7i)T^{2} \) |
| 19 | \( 1 + (1.01e3 + 6.72e3i)T + (-4.49e7 + 1.38e7i)T^{2} \) |
| 23 | \( 1 + (-4.62e3 - 1.17e4i)T + (-1.08e8 + 1.00e8i)T^{2} \) |
| 29 | \( 1 + (2.62e4 - 1.96e3i)T + (5.88e8 - 8.86e7i)T^{2} \) |
| 31 | \( 1 + (-3.42e4 + 2.33e4i)T + (3.24e8 - 8.26e8i)T^{2} \) |
| 37 | \( 1 + (-3.74e4 - 2.15e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (1.79e4 + 8.63e3i)T + (2.96e9 + 3.71e9i)T^{2} \) |
| 47 | \( 1 + (4.49e4 + 5.63e4i)T + (-2.39e9 + 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-1.16e5 - 1.07e5i)T + (1.65e9 + 2.21e10i)T^{2} \) |
| 59 | \( 1 + (-6.44e4 + 2.82e5i)T + (-3.80e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-1.54e5 + 2.26e5i)T + (-1.88e10 - 4.79e10i)T^{2} \) |
| 67 | \( 1 + (-6.05e4 + 9.13e3i)T + (8.64e10 - 2.66e10i)T^{2} \) |
| 71 | \( 1 + (3.96e5 + 1.55e5i)T + (9.39e10 + 8.71e10i)T^{2} \) |
| 73 | \( 1 + (-4.55e5 - 4.91e5i)T + (-1.13e10 + 1.50e11i)T^{2} \) |
| 79 | \( 1 + (4.73e5 + 8.20e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-3.63e4 + 4.84e5i)T + (-3.23e11 - 4.87e10i)T^{2} \) |
| 89 | \( 1 + (8.63e5 + 6.47e4i)T + (4.91e11 + 7.40e10i)T^{2} \) |
| 97 | \( 1 + (-5.59e5 + 7.01e5i)T + (-1.85e11 - 8.12e11i)T^{2} \) |
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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
−15.11386848059297662445257609617, −14.29864100913997490666601048881, −13.15680250196474458865487034708, −11.46189702679204926687922122442, −9.500168586562033909451271352765, −8.792405938595609693972108200511, −7.57221167030264422025255402666, −6.62700906220728309809854803718, −3.58290512116074948853594888933, −2.63352936354875951413614602991,
1.23379024925025662999521916552, 2.62594764541488429107341985623, 4.10955362189991017080327856704, 6.94852564415919409088418590289, 8.393804329828996794243431321462, 9.454604624315650668778825828775, 10.14525271554728826224867948227, 12.27929708807064006590406449816, 12.91881264333582561906195972416, 14.49312397350898980249289847435
