| L(s) = 1 | + (−0.644 − 1.89i)2-s + (1.35 + 1.07i)3-s + (−3.17 + 2.43i)4-s + (−2.18 − 1.05i)5-s + (1.17 − 3.25i)6-s + (1.46 + 1.17i)7-s + (6.66 + 4.43i)8-s + (0.667 + 2.92i)9-s + (−0.585 + 4.81i)10-s + (−12.2 − 2.79i)11-s + (−6.92 − 0.120i)12-s + (4.58 − 20.0i)13-s + (1.27 − 3.53i)14-s + (−1.82 − 3.78i)15-s + (4.10 − 15.4i)16-s − 4.40·17-s + ⋯ |
| L(s) = 1 | + (−0.322 − 0.946i)2-s + (0.451 + 0.359i)3-s + (−0.792 + 0.609i)4-s + (−0.437 − 0.210i)5-s + (0.195 − 0.543i)6-s + (0.209 + 0.167i)7-s + (0.832 + 0.553i)8-s + (0.0741 + 0.324i)9-s + (−0.0585 + 0.481i)10-s + (−1.11 − 0.254i)11-s + (−0.577 − 0.0100i)12-s + (0.352 − 1.54i)13-s + (0.0908 − 0.252i)14-s + (−0.121 − 0.252i)15-s + (0.256 − 0.966i)16-s − 0.259·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.433i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.900 + 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.187624 - 0.821924i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.187624 - 0.821924i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.644 + 1.89i)T \) |
| 3 | \( 1 + (-1.35 - 1.07i)T \) |
| 29 | \( 1 + (28.9 + 2.26i)T \) |
| good | 5 | \( 1 + (2.18 + 1.05i)T + (15.5 + 19.5i)T^{2} \) |
| 7 | \( 1 + (-1.46 - 1.17i)T + (10.9 + 47.7i)T^{2} \) |
| 11 | \( 1 + (12.2 + 2.79i)T + (109. + 52.4i)T^{2} \) |
| 13 | \( 1 + (-4.58 + 20.0i)T + (-152. - 73.3i)T^{2} \) |
| 17 | \( 1 + 4.40T + 289T^{2} \) |
| 19 | \( 1 + (-9.42 + 7.52i)T + (80.3 - 351. i)T^{2} \) |
| 23 | \( 1 + (7.51 + 15.5i)T + (-329. + 413. i)T^{2} \) |
| 31 | \( 1 + (-12.8 + 26.7i)T + (-599. - 751. i)T^{2} \) |
| 37 | \( 1 + (5.76 + 25.2i)T + (-1.23e3 + 593. i)T^{2} \) |
| 41 | \( 1 + 26.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-24.0 - 49.8i)T + (-1.15e3 + 1.44e3i)T^{2} \) |
| 47 | \( 1 + (-55.1 - 12.5i)T + (1.99e3 + 958. i)T^{2} \) |
| 53 | \( 1 + (62.5 + 30.1i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 + 46.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + (61.8 - 77.5i)T + (-828. - 3.62e3i)T^{2} \) |
| 67 | \( 1 + (41.8 - 9.56i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (-28.3 - 6.47i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (-71.5 + 34.4i)T + (3.32e3 - 4.16e3i)T^{2} \) |
| 79 | \( 1 + (-30.9 + 7.05i)T + (5.62e3 - 2.70e3i)T^{2} \) |
| 83 | \( 1 + (61.7 - 49.2i)T + (1.53e3 - 6.71e3i)T^{2} \) |
| 89 | \( 1 + (40.0 + 19.2i)T + (4.93e3 + 6.19e3i)T^{2} \) |
| 97 | \( 1 + (-69.0 - 86.5i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−10.77388392333368978195074150141, −10.13295276364072331740329219933, −9.095926839658642056303240918999, −8.080364737056275135942371894736, −7.76874402118878417249644885388, −5.63421557588142575188093439144, −4.58300063559959677084680918680, −3.39558231862302207072656412493, −2.39883246559927639312756823293, −0.40857475438369213933690044293,
1.71276694095329933151459504067, 3.66123771631077786895150235982, 4.83995955456126288654085751240, 6.07516981704114598693503989519, 7.23751403972560032939735365184, 7.67414365948159025592837796421, 8.731745012399367967226392930064, 9.531302477991545118540477402852, 10.59882345728704658978442889296, 11.62723904533657364983467853332
