| L(s) = 1 | + (−0.381 + 1.36i)2-s + (1.51 + 0.842i)3-s + (−1.70 − 1.03i)4-s + (−1.15 + 3.16i)5-s + (−1.72 + 1.73i)6-s + (−0.984 − 0.173i)7-s + (2.06 − 1.92i)8-s + (1.58 + 2.54i)9-s + (−3.86 − 2.77i)10-s + (−2.52 + 0.919i)11-s + (−1.71 − 3.01i)12-s + (1.75 − 1.47i)13-s + (0.612 − 1.27i)14-s + (−4.40 + 3.81i)15-s + (1.83 + 3.55i)16-s + (−4.09 + 2.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.270 + 0.962i)2-s + (0.873 + 0.486i)3-s + (−0.854 − 0.519i)4-s + (−0.514 + 1.41i)5-s + (−0.704 + 0.710i)6-s + (−0.372 − 0.0656i)7-s + (0.731 − 0.682i)8-s + (0.527 + 0.849i)9-s + (−1.22 − 0.877i)10-s + (−0.761 + 0.277i)11-s + (−0.493 − 0.869i)12-s + (0.485 − 0.407i)13-s + (0.163 − 0.340i)14-s + (−1.13 + 0.985i)15-s + (0.459 + 0.888i)16-s + (−0.992 + 0.572i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 + 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.310928 - 0.901112i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.310928 - 0.901112i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.381 - 1.36i)T \) |
| 3 | \( 1 + (-1.51 - 0.842i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| good | 5 | \( 1 + (1.15 - 3.16i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (2.52 - 0.919i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.75 + 1.47i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (4.09 - 2.36i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.50 + 2.02i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.542 - 3.07i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.207 - 0.247i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-2.34 + 0.414i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.29 + 2.23i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.06 - 6.04i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.74 + 10.2i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.41 + 8.03i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 2.58iT - 53T^{2} \) |
| 59 | \( 1 + (1.37 + 0.499i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (2.54 - 14.4i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.59 - 10.2i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.13 + 1.96i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.35 - 12.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.624 + 0.744i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.80 - 5.70i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (13.1 + 7.57i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.09 - 2.21i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.46551704558620725480517463607, −10.09744328728825583566999068342, −8.907815850010307202141293917033, −8.274397866921402691501945117668, −7.33860166835238630429742687371, −6.83684299726856079136504558048, −5.71595504494182495901780124916, −4.39242070314056290095188266277, −3.56854183562806563223509429567, −2.41656352661344274088239823498,
0.47041483577006942145206739860, 1.82976532827008124116485669372, 2.99839259995572766856159577713, 4.14046261715302458741584196962, 4.83074997620013049737769614883, 6.38781088001581977335276730620, 7.76380269238782141741885280442, 8.340169506812251166489341376100, 8.966437309192454295711467775715, 9.548203258978073602952464726845
