| L(s) = 1 | + (1.24 − 0.669i)2-s + (−0.661 − 1.60i)3-s + (1.10 − 1.66i)4-s + (0.851 − 2.34i)5-s + (−1.89 − 1.55i)6-s + (−0.984 − 0.173i)7-s + (0.260 − 2.81i)8-s + (−2.12 + 2.11i)9-s + (−0.504 − 3.48i)10-s + (−3.41 + 1.24i)11-s + (−3.39 − 0.665i)12-s + (2.31 − 1.94i)13-s + (−1.34 + 0.442i)14-s + (−4.31 + 0.184i)15-s + (−1.55 − 3.68i)16-s + (0.475 − 0.274i)17-s + ⋯ |
| L(s) = 1 | + (0.881 − 0.473i)2-s + (−0.381 − 0.924i)3-s + (0.552 − 0.833i)4-s + (0.381 − 1.04i)5-s + (−0.773 − 0.633i)6-s + (−0.372 − 0.0656i)7-s + (0.0922 − 0.995i)8-s + (−0.708 + 0.705i)9-s + (−0.159 − 1.10i)10-s + (−1.02 + 0.374i)11-s + (−0.981 − 0.192i)12-s + (0.642 − 0.539i)13-s + (−0.358 + 0.118i)14-s + (−1.11 + 0.0476i)15-s + (−0.389 − 0.920i)16-s + (0.115 − 0.0665i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.187291 - 2.03926i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.187291 - 2.03926i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.24 + 0.669i)T \) |
| 3 | \( 1 + (0.661 + 1.60i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| good | 5 | \( 1 + (-0.851 + 2.34i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (3.41 - 1.24i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.31 + 1.94i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.475 + 0.274i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.487 - 0.281i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.02 - 5.81i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.67 + 4.37i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.01 + 0.531i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (4.41 + 7.65i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.71 - 5.61i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.62 + 7.21i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.510 - 2.89i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 1.58iT - 53T^{2} \) |
| 59 | \( 1 + (3.89 + 1.41i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.864 + 4.90i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.55 - 4.23i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (4.92 + 8.52i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.59 - 9.69i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.52 + 10.1i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.419 - 0.351i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-2.19 - 1.26i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-17.3 + 6.32i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.17724165993165986139966958894, −9.204244552777353919438054647579, −8.058005497832153217802394647272, −7.20505044806628910289415702191, −6.04482891488813033070020778516, −5.48796762685517802431785445990, −4.68756865324366204453491574668, −3.20477537344918793536176083063, −2.00542568935095199624570763687, −0.816549149189555338674251862070,
2.68516660987121625607252085984, 3.31195387645891334112085304476, 4.49822875712696749521659657064, 5.39618173088535589329135313173, 6.35343919639708950792357763742, 6.75754718325711411272856116295, 8.136874534731350232014983891303, 8.981038044558746094858707155236, 10.33836536149507064415718345410, 10.61506133268101780350688993835
