| L(s) = 1 | + (0.00525 − 0.0230i)2-s + (−0.382 − 1.67i)3-s + (1.80 + 0.867i)4-s + (2.01 − 2.52i)5-s − 0.0406·6-s + 0.0151·7-s + (0.0589 − 0.0738i)8-s + (0.0380 − 0.0183i)9-s + (−0.0475 − 0.0596i)10-s + (−0.884 + 0.425i)11-s + (0.765 − 3.35i)12-s + (1.86 − 2.33i)13-s + (7.93e−5 − 0.000347i)14-s + (−5.00 − 2.40i)15-s + (2.49 + 3.12i)16-s + (0.623 + 0.781i)17-s + ⋯ |
| L(s) = 1 | + (0.00371 − 0.0162i)2-s + (−0.220 − 0.968i)3-s + (0.900 + 0.433i)4-s + (0.900 − 1.12i)5-s − 0.0165·6-s + 0.00570·7-s + (0.0208 − 0.0261i)8-s + (0.0126 − 0.00610i)9-s + (−0.0150 − 0.0188i)10-s + (−0.266 + 0.128i)11-s + (0.220 − 0.967i)12-s + (0.517 − 0.648i)13-s + (2.12e−5 − 9.29e−5i)14-s + (−1.29 − 0.621i)15-s + (0.622 + 0.781i)16-s + (0.151 + 0.189i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.199 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61849 - 1.32167i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61849 - 1.32167i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-0.623 - 0.781i)T \) |
| 43 | \( 1 + (-1.52 - 6.37i)T \) |
| good | 2 | \( 1 + (-0.00525 + 0.0230i)T + (-1.80 - 0.867i)T^{2} \) |
| 3 | \( 1 + (0.382 + 1.67i)T + (-2.70 + 1.30i)T^{2} \) |
| 5 | \( 1 + (-2.01 + 2.52i)T + (-1.11 - 4.87i)T^{2} \) |
| 7 | \( 1 - 0.0151T + 7T^{2} \) |
| 11 | \( 1 + (0.884 - 0.425i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (-1.86 + 2.33i)T + (-2.89 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-1.28 - 0.619i)T + (11.8 + 14.8i)T^{2} \) |
| 23 | \( 1 + (4.57 - 2.20i)T + (14.3 - 17.9i)T^{2} \) |
| 29 | \( 1 + (0.451 - 1.97i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (0.917 - 4.01i)T + (-27.9 - 13.4i)T^{2} \) |
| 37 | \( 1 - 4.44T + 37T^{2} \) |
| 41 | \( 1 + (0.570 - 2.49i)T + (-36.9 - 17.7i)T^{2} \) |
| 47 | \( 1 + (6.19 + 2.98i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (1.04 + 1.31i)T + (-11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (3.16 + 3.96i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (1.81 + 7.96i)T + (-54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 + (-6.58 - 3.16i)T + (41.7 + 52.3i)T^{2} \) |
| 71 | \( 1 + (9.06 + 4.36i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-9.88 + 12.3i)T + (-16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 - 4.83T + 79T^{2} \) |
| 83 | \( 1 + (0.123 + 0.542i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-2.55 - 11.2i)T + (-80.1 + 38.6i)T^{2} \) |
| 97 | \( 1 + (-4.68 + 2.25i)T + (60.4 - 75.8i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.18324155749527143556624370992, −9.370052277948151087899075015143, −8.143483334253197485629293296272, −7.75105761965629886608118784755, −6.53085259992383672652885531490, −5.99482974345927454354390666081, −4.98934249014385257285185513644, −3.44849119795455406190292871802, −2.00794120117865057768621962778, −1.23714405062603144407479848756,
1.87015104053418813972190889493, 2.87645003659976585958478862301, 4.09914810321953618291302347325, 5.39886172817597101619950931368, 6.12608446534587394009158306638, 6.84712732552013038518569362529, 7.81771005813864215381946360686, 9.315359215713178548347638289651, 9.967696498872755792790472827762, 10.49986757717015628146407179442
