| L(s) = 1 | + (0.160 − 1.40i)2-s + (−1.03 − 1.38i)3-s + (−1.94 − 0.450i)4-s + (−1.93 − 0.705i)5-s + (−2.11 + 1.23i)6-s + (0.744 + 0.131i)7-s + (−0.944 + 2.66i)8-s + (−0.856 + 2.87i)9-s + (−1.30 + 2.60i)10-s + (−0.949 − 2.60i)11-s + (1.39 + 3.17i)12-s + (−0.421 − 0.502i)13-s + (0.303 − 1.02i)14-s + (1.02 + 3.41i)15-s + (3.59 + 1.75i)16-s + (−3.79 + 2.18i)17-s + ⋯ |
| L(s) = 1 | + (0.113 − 0.993i)2-s + (−0.597 − 0.801i)3-s + (−0.974 − 0.225i)4-s + (−0.866 − 0.315i)5-s + (−0.864 + 0.503i)6-s + (0.281 + 0.0496i)7-s + (−0.333 + 0.942i)8-s + (−0.285 + 0.958i)9-s + (−0.411 + 0.825i)10-s + (−0.286 − 0.786i)11-s + (0.401 + 0.915i)12-s + (−0.116 − 0.139i)13-s + (0.0811 − 0.273i)14-s + (0.265 + 0.882i)15-s + (0.898 + 0.438i)16-s + (−0.919 + 0.531i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.175858 + 0.449706i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.175858 + 0.449706i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.160 + 1.40i)T \) |
| 3 | \( 1 + (1.03 + 1.38i)T \) |
| good | 5 | \( 1 + (1.93 + 0.705i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.744 - 0.131i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (0.949 + 2.60i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (0.421 + 0.502i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.79 - 2.18i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.155 + 0.270i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.32 + 7.53i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.89 + 4.94i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.16 + 0.733i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (2.70 - 1.56i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.31 - 5.14i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.00 - 1.09i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.89 + 10.7i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 0.876T + 53T^{2} \) |
| 59 | \( 1 + (-2.94 + 8.09i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-12.1 - 2.14i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.32 + 2.79i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-7.08 - 12.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.17 + 12.4i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.09 - 8.44i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.10 - 8.47i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-5.55 - 3.20i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.59 + 0.579i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.58950186473944942789766262926, −11.19495493453589955399556126348, −10.12799387666696581316784690457, −8.475413264777976124148938551530, −8.083464720798763565018172057788, −6.43920591424961858873992540572, −5.17716448930627153020756173915, −4.04552292121064243720889336463, −2.30617536131674085754492037859, −0.42024481720858237234490238468,
3.62176149216634885910735447432, 4.59293713034763704531595808439, 5.55694435048434855775798637685, 6.92362246922317838881509181249, 7.67396272374395496392196099293, 8.997861887379816089388539042948, 9.833829975834226701977127614357, 11.05488641871048333838893290183, 11.86367588093204015296403695975, 12.92262341204070662301351166650
