| L(s) = 1 | + (6.45 + 8.10i)2-s + (−16.9 + 21.2i)3-s + (−16.7 + 73.4i)4-s + (36.1 − 17.4i)5-s − 281.·6-s + 237.·7-s + (−404. + 194. i)8-s + (−110. − 482. i)9-s + (374. + 180. i)10-s + (−32.8 − 144. i)11-s + (−1.27e3 − 1.59e3i)12-s + (−470. + 226. i)13-s + (1.53e3 + 1.92e3i)14-s + (−242. + 1.06e3i)15-s + (−2.01e3 − 972. i)16-s + (1.13e3 + 547. i)17-s + ⋯ |
| L(s) = 1 | + (1.14 + 1.43i)2-s + (−1.08 + 1.36i)3-s + (−0.523 + 2.29i)4-s + (0.646 − 0.311i)5-s − 3.19·6-s + 1.83·7-s + (−2.23 + 1.07i)8-s + (−0.453 − 1.98i)9-s + (1.18 + 0.570i)10-s + (−0.0819 − 0.359i)11-s + (−2.55 − 3.20i)12-s + (−0.771 + 0.371i)13-s + (2.09 + 2.62i)14-s + (−0.278 + 1.21i)15-s + (−1.97 − 0.949i)16-s + (0.953 + 0.459i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.302490 - 2.37721i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.302490 - 2.37721i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.11e4 + 4.77e3i)T \) |
| good | 2 | \( 1 + (-6.45 - 8.10i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (16.9 - 21.2i)T + (-54.0 - 236. i)T^{2} \) |
| 5 | \( 1 + (-36.1 + 17.4i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 - 237.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (32.8 + 144. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (470. - 226. i)T + (2.31e5 - 2.90e5i)T^{2} \) |
| 17 | \( 1 + (-1.13e3 - 547. i)T + (8.85e5 + 1.11e6i)T^{2} \) |
| 19 | \( 1 + (-347. + 1.52e3i)T + (-2.23e6 - 1.07e6i)T^{2} \) |
| 23 | \( 1 + (-280. - 1.23e3i)T + (-5.79e6 + 2.79e6i)T^{2} \) |
| 29 | \( 1 + (2.61e3 + 3.27e3i)T + (-4.56e6 + 1.99e7i)T^{2} \) |
| 31 | \( 1 + (2.74e3 + 3.44e3i)T + (-6.37e6 + 2.79e7i)T^{2} \) |
| 37 | \( 1 - 4.49e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (122. + 153. i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (652. - 2.85e3i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (-1.04e4 - 5.00e3i)T + (2.60e8 + 3.26e8i)T^{2} \) |
| 59 | \( 1 + (1.58e4 + 7.61e3i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (2.66e4 - 3.34e4i)T + (-1.87e8 - 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-5.80e3 + 2.54e4i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-8.66e3 + 3.79e4i)T + (-1.62e9 - 7.82e8i)T^{2} \) |
| 73 | \( 1 + (-3.33e4 + 1.60e4i)T + (1.29e9 - 1.62e9i)T^{2} \) |
| 79 | \( 1 - 6.36e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + (4.72e4 - 5.92e4i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (3.88e3 - 4.87e3i)T + (-1.24e9 - 5.44e9i)T^{2} \) |
| 97 | \( 1 + (-2.56e4 - 1.12e5i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
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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
−15.36542920687339861944932116268, −14.78135028271277725933674841965, −13.72217055000979176132558522692, −12.06618929417731679992700113442, −11.07935103056246929878669948239, −9.294576608328385760319489538975, −7.67690270591312450050877552860, −5.76331441580933046701878454429, −5.16322890670570577843950089654, −4.21115957993382679765796496161,
1.21665265369675820918986172346, 2.18811045947755684331037022052, 4.95424630101547833884522927331, 5.74737147123389624057956927220, 7.58662462748948031593381980761, 10.24570764252045268967577043559, 11.19500782109128614012111726695, 12.04919422626006815980974863290, 12.73486082309480582145496877993, 14.10180869858372678390723310799
