| L(s) = 1 | + (3.22 − 6.69i)2-s + (−59.4 − 123. i)3-s + (125. + 156. i)4-s + (267. − 61.0i)5-s − 1.01e3·6-s − 4.43e3i·7-s + (3.31e3 − 755. i)8-s + (−7.62e3 + 9.55e3i)9-s + (453. − 1.98e3i)10-s + (9.75e3 − 1.22e4i)11-s + (1.19e4 − 2.47e4i)12-s + (−6.42e3 − 2.81e4i)13-s + (−2.97e4 − 1.43e4i)14-s + (−2.34e4 − 2.93e4i)15-s + (−5.81e3 + 2.54e4i)16-s + (−2.80e4 + 1.23e5i)17-s + ⋯ |
| L(s) = 1 | + (0.201 − 0.418i)2-s + (−0.734 − 1.52i)3-s + (0.488 + 0.613i)4-s + (0.427 − 0.0976i)5-s − 0.786·6-s − 1.84i·7-s + (0.808 − 0.184i)8-s + (−1.16 + 1.45i)9-s + (0.0453 − 0.198i)10-s + (0.666 − 0.835i)11-s + (0.575 − 1.19i)12-s + (−0.225 − 0.985i)13-s + (−0.774 − 0.372i)14-s + (−0.462 − 0.580i)15-s + (−0.0887 + 0.388i)16-s + (−0.336 + 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0620i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.998 - 0.0620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0528905 + 1.70207i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0528905 + 1.70207i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.84e6 - 1.90e6i)T \) |
| good | 2 | \( 1 + (-3.22 + 6.69i)T + (-159. - 200. i)T^{2} \) |
| 3 | \( 1 + (59.4 + 123. i)T + (-4.09e3 + 5.12e3i)T^{2} \) |
| 5 | \( 1 + (-267. + 61.0i)T + (3.51e5 - 1.69e5i)T^{2} \) |
| 7 | \( 1 + 4.43e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (-9.75e3 + 1.22e4i)T + (-4.76e7 - 2.08e8i)T^{2} \) |
| 13 | \( 1 + (6.42e3 + 2.81e4i)T + (-7.34e8 + 3.53e8i)T^{2} \) |
| 17 | \( 1 + (2.80e4 - 1.23e5i)T + (-6.28e9 - 3.02e9i)T^{2} \) |
| 19 | \( 1 + (3.27e4 - 2.61e4i)T + (3.77e9 - 1.65e10i)T^{2} \) |
| 23 | \( 1 + (-7.97e4 + 1.00e5i)T + (-1.74e10 - 7.63e10i)T^{2} \) |
| 29 | \( 1 + (1.82e5 - 3.78e5i)T + (-3.11e11 - 3.91e11i)T^{2} \) |
| 31 | \( 1 + (4.47e5 + 2.15e5i)T + (5.31e11 + 6.66e11i)T^{2} \) |
| 37 | \( 1 + 3.20e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (-1.45e6 - 6.98e5i)T + (4.97e12 + 6.24e12i)T^{2} \) |
| 47 | \( 1 + (2.10e6 + 2.64e6i)T + (-5.29e12 + 2.32e13i)T^{2} \) |
| 53 | \( 1 + (8.71e4 - 3.81e5i)T + (-5.60e13 - 2.70e13i)T^{2} \) |
| 59 | \( 1 + (-3.76e6 + 1.64e7i)T + (-1.32e14 - 6.37e13i)T^{2} \) |
| 61 | \( 1 + (-8.30e6 - 1.72e7i)T + (-1.19e14 + 1.49e14i)T^{2} \) |
| 67 | \( 1 + (1.58e6 + 1.98e6i)T + (-9.03e13 + 3.95e14i)T^{2} \) |
| 71 | \( 1 + (-5.07e6 + 4.05e6i)T + (1.43e14 - 6.29e14i)T^{2} \) |
| 73 | \( 1 + (-4.41e7 + 1.00e7i)T + (7.26e14 - 3.49e14i)T^{2} \) |
| 79 | \( 1 + 1.21e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (-1.40e7 + 6.75e6i)T + (1.40e15 - 1.76e15i)T^{2} \) |
| 89 | \( 1 + (3.66e7 + 7.61e7i)T + (-2.45e15 + 3.07e15i)T^{2} \) |
| 97 | \( 1 + (-6.71e7 + 8.42e7i)T + (-1.74e15 - 7.64e15i)T^{2} \) |
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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
−13.08045037189967574765907891423, −12.76534358663523320155101990948, −11.23870477137217183015287154674, −10.61131034367546271089061876642, −8.025592622482737523058503009543, −7.13971970815260448817897613236, −6.06001297310945083332241129500, −3.76681194967174090359285710474, −1.76070120764871798145380500607, −0.65261388926598661166554242459,
2.26856165067421472677748374563, 4.64886898781942979587435507264, 5.52960317689371001171472429398, 6.59472414133705097895551945530, 9.261310063446534809187106134237, 9.730347294010994382528332197435, 11.31962450125008742680741348735, 11.92897914333034906692469008899, 14.20412233391452325096689408567, 15.20584922111058617962511881291
