| L(s) = 1 | + (−0.752 − 2.06i)2-s + (1.63 + 0.559i)3-s + (−2.17 + 1.82i)4-s + (−0.562 − 3.19i)5-s + (−0.0757 − 3.80i)6-s + (2.51 + 0.815i)7-s + (1.59 + 0.923i)8-s + (2.37 + 1.83i)9-s + (−6.17 + 3.56i)10-s + (−3.18 − 0.560i)11-s + (−4.58 + 1.77i)12-s + (1.22 − 3.37i)13-s + (−0.206 − 5.81i)14-s + (0.864 − 5.54i)15-s + (−0.280 + 1.59i)16-s + (1.49 + 2.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.532 − 1.46i)2-s + (0.946 + 0.323i)3-s + (−1.08 + 0.912i)4-s + (−0.251 − 1.42i)5-s + (−0.0309 − 1.55i)6-s + (0.951 + 0.308i)7-s + (0.565 + 0.326i)8-s + (0.790 + 0.611i)9-s + (−1.95 + 1.12i)10-s + (−0.958 − 0.169i)11-s + (−1.32 + 0.512i)12-s + (0.340 − 0.935i)13-s + (−0.0553 − 1.55i)14-s + (0.223 − 1.43i)15-s + (−0.0701 + 0.397i)16-s + (0.361 + 0.626i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.511683 - 1.04629i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.511683 - 1.04629i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.63 - 0.559i)T \) |
| 7 | \( 1 + (-2.51 - 0.815i)T \) |
| good | 2 | \( 1 + (0.752 + 2.06i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (0.562 + 3.19i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (3.18 + 0.560i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.22 + 3.37i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.49 - 2.58i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.49 + 2.01i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.441 + 0.526i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.33 - 9.15i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.311 + 0.371i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.82 - 4.88i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.84 - 0.671i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.32 + 7.48i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-6.71 - 5.63i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 3.11iT - 53T^{2} \) |
| 59 | \( 1 + (1.95 + 11.0i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (3.09 - 3.68i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (8.93 + 3.25i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (2.67 - 1.54i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-9.91 - 5.72i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.20 + 2.25i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (16.4 - 6.00i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (3.48 - 6.03i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.47 + 0.436i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.44983525407561128407837867243, −10.96626132323274320342955247439, −10.39179386398005830049108287818, −9.136021438014071096263668483705, −8.463137379700984957964184110896, −7.988857856875720046177089885138, −5.23821682704542082642076989014, −4.20006219892895726048691481263, −2.77194811220036116342126566942, −1.34827219096548597625007484269,
2.51891853081436491611115526079, 4.29353507404943199666342446200, 6.06967306343718298943910742487, 7.12311553609342685590795158346, 7.67493331655990207980439048644, 8.418361652079831986495411989433, 9.649086910012493302839276906213, 10.68240970961557806140644387733, 11.86948005530801995594144806337, 13.58784525474889067215088527143
