| L(s) = 1 | + (−1.03 + 0.813i)2-s + (0.814 − 0.580i)3-s + (−0.0634 + 0.261i)4-s + (−0.255 − 0.0493i)5-s + (−0.370 + 1.26i)6-s + (−2.61 − 0.373i)7-s + (−1.24 − 2.71i)8-s + (0.327 − 0.945i)9-s + (0.304 − 0.157i)10-s + (1.25 − 1.59i)11-s + (0.0999 + 0.249i)12-s + (1.10 − 1.71i)13-s + (3.01 − 1.74i)14-s + (−0.237 + 0.108i)15-s + (3.01 + 1.55i)16-s + (5.65 − 5.38i)17-s + ⋯ |
| L(s) = 1 | + (−0.731 + 0.575i)2-s + (0.470 − 0.334i)3-s + (−0.0317 + 0.130i)4-s + (−0.114 − 0.0220i)5-s + (−0.151 + 0.515i)6-s + (−0.990 − 0.141i)7-s + (−0.438 − 0.960i)8-s + (0.109 − 0.315i)9-s + (0.0963 − 0.0496i)10-s + (0.377 − 0.480i)11-s + (0.0288 + 0.0720i)12-s + (0.305 − 0.475i)13-s + (0.805 − 0.466i)14-s + (−0.0612 + 0.0279i)15-s + (0.753 + 0.388i)16-s + (1.37 − 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.865262 - 0.227670i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.865262 - 0.227670i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.814 + 0.580i)T \) |
| 7 | \( 1 + (2.61 + 0.373i)T \) |
| 23 | \( 1 + (-4.54 - 1.52i)T \) |
| good | 2 | \( 1 + (1.03 - 0.813i)T + (0.471 - 1.94i)T^{2} \) |
| 5 | \( 1 + (0.255 + 0.0493i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (-1.25 + 1.59i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-1.10 + 1.71i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-5.65 + 5.38i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.566 - 0.539i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (-2.59 - 0.761i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.0699 + 0.732i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (9.20 + 3.18i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (5.57 + 4.82i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-10.1 - 4.63i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-8.06 + 4.65i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.40 - 0.162i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (6.10 + 11.8i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (5.30 - 7.44i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (-4.50 + 11.2i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (0.470 + 3.27i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-7.87 - 1.91i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (9.01 + 0.429i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-10.8 - 12.5i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-5.44 + 0.519i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (8.55 - 9.86i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.64003825310654864060570374246, −9.580217734369022019849765394842, −9.151173020151012451947454172379, −8.116842039795522440424314703429, −7.38729442362893610096932166577, −6.64422895488311481807206665836, −5.56224927555834101466302994978, −3.70401973871562770505113069700, −3.04288703489811368286750558483, −0.71936339692783610782536417333,
1.47364453260071407178467151673, 2.90752089608566299599173868996, 3.97081985555463168899743798984, 5.43244516135589436895349743384, 6.44580551529739589752721322585, 7.69420397172928115840018801470, 8.768523292701429271975943361463, 9.317312372140380075516821959407, 10.15503507794001688034596641141, 10.64886052297506693832045067214
