| L(s) = 1 | + (0.483 − 1.32i)2-s + (−0.173 + 0.0305i)3-s + (−1.53 − 1.28i)4-s + (6.04 − 5.07i)5-s + (−0.0431 + 0.244i)6-s + (−3.82 + 6.63i)7-s + (−2.44 + 1.41i)8-s + (−8.42 + 3.06i)9-s + (−3.81 − 10.4i)10-s + (7.09 + 12.2i)11-s + (0.304 + 0.175i)12-s + (6.69 + 1.18i)13-s + (6.96 + 8.29i)14-s + (−0.891 + 1.06i)15-s + (0.694 + 3.93i)16-s + (−16.4 − 5.99i)17-s + ⋯ |
| L(s) = 1 | + (0.241 − 0.664i)2-s + (−0.0577 + 0.0101i)3-s + (−0.383 − 0.321i)4-s + (1.20 − 1.01i)5-s + (−0.00719 + 0.0408i)6-s + (−0.547 + 0.947i)7-s + (−0.306 + 0.176i)8-s + (−0.936 + 0.340i)9-s + (−0.381 − 1.04i)10-s + (0.644 + 1.11i)11-s + (0.0253 + 0.0146i)12-s + (0.515 + 0.0908i)13-s + (0.497 + 0.592i)14-s + (−0.0594 + 0.0708i)15-s + (0.0434 + 0.246i)16-s + (−0.968 − 0.352i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.06152 - 0.548724i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06152 - 0.548724i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.483 + 1.32i)T \) |
| 19 | \( 1 + (4.02 + 18.5i)T \) |
| good | 3 | \( 1 + (0.173 - 0.0305i)T + (8.45 - 3.07i)T^{2} \) |
| 5 | \( 1 + (-6.04 + 5.07i)T + (4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (3.82 - 6.63i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-7.09 - 12.2i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-6.69 - 1.18i)T + (158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (16.4 + 5.99i)T + (221. + 185. i)T^{2} \) |
| 23 | \( 1 + (-4.48 - 3.76i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-3.32 - 9.13i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (11.0 + 6.40i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 64.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-60.3 + 10.6i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (42.7 - 35.8i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (45.2 - 16.4i)T + (1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (39.5 - 47.1i)T + (-487. - 2.76e3i)T^{2} \) |
| 59 | \( 1 + (-23.9 + 65.8i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-24.0 - 20.2i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (30.0 + 82.6i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-34.7 - 41.3i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-4.30 - 24.4i)T + (-5.00e3 + 1.82e3i)T^{2} \) |
| 79 | \( 1 + (37.9 - 6.69i)T + (5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-23.7 + 41.1i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-108. - 19.0i)T + (7.44e3 + 2.70e3i)T^{2} \) |
| 97 | \( 1 + (14.4 - 39.5i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
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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
−15.95619977102620167562941650931, −14.46829611207238689374885913981, −13.25735227891018954485190807307, −12.50814034466082819269293271583, −11.17470116163876143097261746736, −9.395551166229909241112574358607, −8.982201658981329123491366707939, −6.16322034334647332006910956627, −4.91098669654766318424589924998, −2.24567151131638886747146423304,
3.42241894168030790342498892381, 6.02149407688509036198057588142, 6.61799521309331439090430822242, 8.615406481148617672091992350140, 10.08199670158794367252455605641, 11.26593478692287385481199569578, 13.28700368478413662724038497516, 13.96980250776526249103551645284, 14.81426360679025699194218318836, 16.49034314764920261088988391730
