| L(s) = 1 | + (−0.857 − 0.329i)2-s + (−2.24 + 0.235i)3-s + (−0.858 − 0.773i)4-s + (1.02 + 1.02i)5-s + (2.00 + 0.536i)6-s + (0.00452 + 0.0863i)7-s + (1.31 + 2.58i)8-s + (2.04 − 0.434i)9-s + (−0.543 − 1.21i)10-s + (0.549 + 0.845i)11-s + (2.10 + 1.53i)12-s + (−3.52 − 0.762i)13-s + (0.0245 − 0.0755i)14-s + (−2.54 − 2.06i)15-s + (−0.0369 − 0.351i)16-s + (6.67 − 1.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.606 − 0.232i)2-s + (−1.29 + 0.136i)3-s + (−0.429 − 0.386i)4-s + (0.459 + 0.459i)5-s + (0.817 + 0.219i)6-s + (0.00170 + 0.0326i)7-s + (0.465 + 0.913i)8-s + (0.681 − 0.144i)9-s + (−0.171 − 0.385i)10-s + (0.165 + 0.254i)11-s + (0.609 + 0.442i)12-s + (−0.977 − 0.211i)13-s + (0.00655 − 0.0201i)14-s + (−0.657 − 0.532i)15-s + (−0.00922 − 0.0878i)16-s + (1.61 − 0.344i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.257687 - 0.303805i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.257687 - 0.303805i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (3.52 + 0.762i)T \) |
| 31 | \( 1 + (4.94 + 2.55i)T \) |
| good | 2 | \( 1 + (0.857 + 0.329i)T + (1.48 + 1.33i)T^{2} \) |
| 3 | \( 1 + (2.24 - 0.235i)T + (2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + (-1.02 - 1.02i)T + 5iT^{2} \) |
| 7 | \( 1 + (-0.00452 - 0.0863i)T + (-6.96 + 0.731i)T^{2} \) |
| 11 | \( 1 + (-0.549 - 0.845i)T + (-4.47 + 10.0i)T^{2} \) |
| 17 | \( 1 + (-6.67 + 1.41i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (0.975 - 0.790i)T + (3.95 - 18.5i)T^{2} \) |
| 23 | \( 1 + (4.08 + 4.53i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (3.28 + 7.38i)T + (-19.4 + 21.5i)T^{2} \) |
| 37 | \( 1 + (-5.17 + 1.38i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (9.39 + 3.60i)T + (30.4 + 27.4i)T^{2} \) |
| 43 | \( 1 + (0.128 - 1.21i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (-0.810 + 5.11i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-13.4 - 4.36i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (9.43 - 3.62i)T + (43.8 - 39.4i)T^{2} \) |
| 61 | \( 1 + (-8.11 + 4.68i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.64 + 0.709i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (0.733 + 0.476i)T + (28.8 + 64.8i)T^{2} \) |
| 73 | \( 1 + (0.975 - 1.91i)T + (-42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (-0.390 - 0.126i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-5.96 + 0.944i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (5.07 + 7.81i)T + (-36.1 + 81.3i)T^{2} \) |
| 97 | \( 1 + (13.0 - 0.682i)T + (96.4 - 10.1i)T^{2} \) |
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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
−10.76488648626513710246897499460, −10.06654336804722569119463463005, −9.727976645897775342142612719480, −8.325151430509992849944440669860, −7.23938673338390924494828403551, −5.93980843176285930856979965438, −5.45130585151059610583743970521, −4.30467573464692432050251584326, −2.25152177143842478936373213789, −0.42982374892395942846542694711,
1.25520978446637314378859277896, 3.61851410121310223869831242492, 5.05666232397558033072557285590, 5.65534823897607709277420275124, 6.92135034606550925526062280434, 7.72627494023924278751779370442, 8.875225534788326799809068811726, 9.716777752954213663398565979173, 10.43394912790608229640032318335, 11.60631288253443904523967115991
