| L(s) = 1 | + (0.186 − 0.234i)2-s + (−17.0 − 21.4i)3-s + (7.10 + 31.1i)4-s + (−36.3 − 17.4i)5-s − 8.19·6-s + 50.2·7-s + (17.2 + 8.30i)8-s + (−112. + 493. i)9-s + (−10.8 + 5.23i)10-s + (−100. + 441. i)11-s + (544. − 682. i)12-s + (894. + 430. i)13-s + (9.38 − 11.7i)14-s + (245. + 1.07e3i)15-s + (−914. + 440. i)16-s + (−1.43e3 + 688. i)17-s + ⋯ |
| L(s) = 1 | + (0.0329 − 0.0413i)2-s + (−1.09 − 1.37i)3-s + (0.221 + 0.972i)4-s + (−0.649 − 0.312i)5-s − 0.0929·6-s + 0.387·7-s + (0.0952 + 0.0458i)8-s + (−0.463 + 2.03i)9-s + (−0.0343 + 0.0165i)10-s + (−0.250 + 1.09i)11-s + (1.09 − 1.36i)12-s + (1.46 + 0.707i)13-s + (0.0127 − 0.0160i)14-s + (0.281 + 1.23i)15-s + (−0.893 + 0.430i)16-s + (−1.20 + 0.578i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0894 - 0.995i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0894 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.442367 + 0.404426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.442367 + 0.404426i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (2.89e3 - 1.17e4i)T \) |
| good | 2 | \( 1 + (-0.186 + 0.234i)T + (-7.12 - 31.1i)T^{2} \) |
| 3 | \( 1 + (17.0 + 21.4i)T + (-54.0 + 236. i)T^{2} \) |
| 5 | \( 1 + (36.3 + 17.4i)T + (1.94e3 + 2.44e3i)T^{2} \) |
| 7 | \( 1 - 50.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + (100. - 441. i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-894. - 430. i)T + (2.31e5 + 2.90e5i)T^{2} \) |
| 17 | \( 1 + (1.43e3 - 688. i)T + (8.85e5 - 1.11e6i)T^{2} \) |
| 19 | \( 1 + (308. + 1.35e3i)T + (-2.23e6 + 1.07e6i)T^{2} \) |
| 23 | \( 1 + (512. - 2.24e3i)T + (-5.79e6 - 2.79e6i)T^{2} \) |
| 29 | \( 1 + (3.69e3 - 4.63e3i)T + (-4.56e6 - 1.99e7i)T^{2} \) |
| 31 | \( 1 + (-3.75e3 + 4.71e3i)T + (-6.37e6 - 2.79e7i)T^{2} \) |
| 37 | \( 1 + 729.T + 6.93e7T^{2} \) |
| 41 | \( 1 + (6.48e3 - 8.12e3i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 47 | \( 1 + (-2.61e3 - 1.14e4i)T + (-2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-1.69e4 + 8.17e3i)T + (2.60e8 - 3.26e8i)T^{2} \) |
| 59 | \( 1 + (-4.06e3 + 1.95e3i)T + (4.45e8 - 5.58e8i)T^{2} \) |
| 61 | \( 1 + (-1.02e4 - 1.28e4i)T + (-1.87e8 + 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-2.93e3 - 1.28e4i)T + (-1.21e9 + 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-4.20e3 - 1.84e4i)T + (-1.62e9 + 7.82e8i)T^{2} \) |
| 73 | \( 1 + (5.74e4 + 2.76e4i)T + (1.29e9 + 1.62e9i)T^{2} \) |
| 79 | \( 1 - 7.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (1.99e4 + 2.50e4i)T + (-8.76e8 + 3.84e9i)T^{2} \) |
| 89 | \( 1 + (7.77e4 + 9.74e4i)T + (-1.24e9 + 5.44e9i)T^{2} \) |
| 97 | \( 1 + (1.42e4 - 6.22e4i)T + (-7.73e9 - 3.72e9i)T^{2} \) |
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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
−15.54815090586984934760753826451, −13.37247538272264664794817529736, −12.83211566037509933982948539038, −11.61662636140262267279605355085, −11.22919478316267941716540324743, −8.506615195883587488603807798625, −7.44640047005053016705140365809, −6.43542193458248673190907358167, −4.43148098851156780172171966682, −1.78747508427080132587383124710,
0.36773611873695580275510870842, 3.85389052213728288408756222457, 5.33834589586572706044369352187, 6.31826143458739582572486907602, 8.656861988837320015736164408889, 10.28338219068792215445719675157, 10.97179559208865440930893709982, 11.59161059214770336764294406100, 13.77197665885488525254719185571, 15.24352196918713218716942106934
