| L(s) = 1 | + (1.52 − 0.230i)2-s + (−0.192 − 0.131i)3-s + (0.373 − 0.115i)4-s + (0.125 − 1.66i)5-s + (−0.325 − 0.156i)6-s + (−2.31 − 1.28i)7-s + (−2.24 + 1.07i)8-s + (−1.07 − 2.74i)9-s + (−0.193 − 2.58i)10-s + (0.365 − 0.930i)11-s + (−0.0872 − 0.0268i)12-s + (−2.04 − 2.56i)13-s + (−3.83 − 1.42i)14-s + (−0.243 + 0.305i)15-s + (−3.82 + 2.60i)16-s + (5.22 − 4.85i)17-s + ⋯ |
| L(s) = 1 | + (1.08 − 0.162i)2-s + (−0.111 − 0.0759i)3-s + (0.186 − 0.0576i)4-s + (0.0559 − 0.746i)5-s + (−0.132 − 0.0639i)6-s + (−0.874 − 0.485i)7-s + (−0.792 + 0.381i)8-s + (−0.358 − 0.913i)9-s + (−0.0611 − 0.816i)10-s + (0.110 − 0.280i)11-s + (−0.0251 − 0.00776i)12-s + (−0.566 − 0.710i)13-s + (−1.02 − 0.382i)14-s + (−0.0628 + 0.0788i)15-s + (−0.956 + 0.651i)16-s + (1.26 − 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.788103 - 1.34519i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.788103 - 1.34519i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.31 + 1.28i)T \) |
| 11 | \( 1 + (-0.365 + 0.930i)T \) |
| good | 2 | \( 1 + (-1.52 + 0.230i)T + (1.91 - 0.589i)T^{2} \) |
| 3 | \( 1 + (0.192 + 0.131i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.125 + 1.66i)T + (-4.94 - 0.745i)T^{2} \) |
| 13 | \( 1 + (2.04 + 2.56i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-5.22 + 4.85i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (1.24 - 2.16i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.84 - 2.63i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (0.811 + 3.55i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-3.09 - 5.35i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (10.2 + 3.16i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-5.51 + 2.65i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-0.853 - 0.410i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-6.24 + 0.941i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (11.9 - 3.68i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.459 + 6.12i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (1.13 + 0.349i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (0.315 + 0.546i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.61 + 11.4i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-14.3 - 2.16i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-2.41 + 4.17i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.27 + 2.84i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (0.458 + 1.16i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + 9.06T + 97T^{2} \) |
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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
−10.63677552495107389416036461126, −9.500302813579602605409016667963, −9.009544561382893729321651591199, −7.73701133548267748459525230199, −6.58251055932098238362994167789, −5.61925691671169892931474174902, −4.93801845897356879172695504985, −3.62034319189019621926794192734, −3.02799891592882133562908126355, −0.64221077374254233472162811389,
2.47303890959194999461021534108, 3.38678944542286320571387005953, 4.57497770751736569870352773519, 5.52140372083622269855032472754, 6.36419849569669229322750567275, 7.11483090190477811395032455175, 8.462748112751643443825121355111, 9.478369088804744603179578356446, 10.29547955153538173938335342643, 11.19874256397254831555942843942
