| L(s) = 1 | + (−6.35 + 13.1i)2-s + (−28.5 + 41.8i)3-s + (−93.6 − 117. i)4-s + (6.53 + 7.04i)5-s + (−370. − 642. i)6-s + (234. + 135. i)7-s + (1.23e3 − 280. i)8-s + (−671. − 1.71e3i)9-s + (−134. + 41.4i)10-s + (−1.49e3 + 1.87e3i)11-s + (7.59e3 − 568. i)12-s + (−2.46e3 − 761. i)13-s + (−3.27e3 + 2.23e3i)14-s + (−481. + 72.5i)15-s + (−1.97e3 + 8.63e3i)16-s + (1.31e3 + 1.22e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.793 + 1.64i)2-s + (−1.05 + 1.55i)3-s + (−1.46 − 1.83i)4-s + (0.0522 + 0.0563i)5-s + (−1.71 − 2.97i)6-s + (0.684 + 0.395i)7-s + (2.40 − 0.548i)8-s + (−0.921 − 2.34i)9-s + (−0.134 + 0.0414i)10-s + (−1.12 + 1.41i)11-s + (4.39 − 0.329i)12-s + (−1.12 − 0.346i)13-s + (−1.19 + 0.814i)14-s + (−0.142 + 0.0214i)15-s + (−0.481 + 2.10i)16-s + (0.267 + 0.248i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0815430 + 0.00807628i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0815430 + 0.00807628i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (7.94e4 - 2.95e3i)T \) |
| good | 2 | \( 1 + (6.35 - 13.1i)T + (-39.9 - 50.0i)T^{2} \) |
| 3 | \( 1 + (28.5 - 41.8i)T + (-266. - 678. i)T^{2} \) |
| 5 | \( 1 + (-6.53 - 7.04i)T + (-1.16e3 + 1.55e4i)T^{2} \) |
| 7 | \( 1 + (-234. - 135. i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (1.49e3 - 1.87e3i)T + (-3.94e5 - 1.72e6i)T^{2} \) |
| 13 | \( 1 + (2.46e3 + 761. i)T + (3.98e6 + 2.71e6i)T^{2} \) |
| 17 | \( 1 + (-1.31e3 - 1.22e3i)T + (1.80e6 + 2.40e7i)T^{2} \) |
| 19 | \( 1 + (-2.37e3 - 931. i)T + (3.44e7 + 3.19e7i)T^{2} \) |
| 23 | \( 1 + (-4.21e3 - 635. i)T + (1.41e8 + 4.36e7i)T^{2} \) |
| 29 | \( 1 + (-1.49e4 - 2.18e4i)T + (-2.17e8 + 5.53e8i)T^{2} \) |
| 31 | \( 1 + (-1.31e3 - 1.76e4i)T + (-8.77e8 + 1.32e8i)T^{2} \) |
| 37 | \( 1 + (3.19e4 - 1.84e4i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + (8.39e4 + 4.04e4i)T + (2.96e9 + 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-9.43e3 - 1.18e4i)T + (-2.39e9 + 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-2.64e5 + 8.17e4i)T + (1.83e10 - 1.24e10i)T^{2} \) |
| 59 | \( 1 + (-3.79e4 + 1.66e5i)T + (-3.80e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-1.49e5 - 1.12e4i)T + (5.09e10 + 7.67e9i)T^{2} \) |
| 67 | \( 1 + (-5.97e4 + 1.52e5i)T + (-6.63e10 - 6.15e10i)T^{2} \) |
| 71 | \( 1 + (7.34e3 + 4.87e4i)T + (-1.22e11 + 3.77e10i)T^{2} \) |
| 73 | \( 1 + (-2.48e4 + 8.04e4i)T + (-1.25e11 - 8.52e10i)T^{2} \) |
| 79 | \( 1 + (1.25e5 - 2.17e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (5.55e5 + 3.78e5i)T + (1.19e11 + 3.04e11i)T^{2} \) |
| 89 | \( 1 + (1.43e5 - 2.09e5i)T + (-1.81e11 - 4.62e11i)T^{2} \) |
| 97 | \( 1 + (-1.80e5 + 2.26e5i)T + (-1.85e11 - 8.12e11i)T^{2} \) |
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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
−15.87061516002831456999689877685, −15.17709379527768878619920410323, −14.57050726474985445532471750871, −12.15105793894405070545872667121, −10.31627670985035319325315570739, −9.939277478485765890701013962065, −8.477669111668260871277858953878, −6.94314253287539266175040483937, −5.22180644007470487356957953678, −4.94581470557018831615992692559,
0.06793021863786863003389950434, 1.13216083318856740717370445919, 2.55741661670586541983443182585, 5.22854544222061443225760595498, 7.40312394544256056582966434733, 8.361631904578489822548629308007, 10.31237629421529955080713438056, 11.33898270806453692431370353957, 11.86337009211895785874933235757, 13.09667041696197543089987502950
