| L(s) = 1 | + (1.03 + 1.23i)2-s + (1.23 + 1.21i)3-s + (−0.105 + 0.597i)4-s + (−0.212 + 2.78i)6-s + (1.71 − 0.302i)7-s + (1.94 − 1.12i)8-s + (0.0676 + 2.99i)9-s + (2.93 + 1.06i)11-s + (−0.853 + 0.612i)12-s + (1.21 − 1.44i)13-s + (2.15 + 1.80i)14-s + (4.55 + 1.65i)16-s + (−5.56 − 3.21i)17-s + (−3.63 + 3.19i)18-s + (−2.43 − 4.21i)19-s + ⋯ |
| L(s) = 1 | + (0.733 + 0.874i)2-s + (0.715 + 0.699i)3-s + (−0.0526 + 0.298i)4-s + (−0.0866 + 1.13i)6-s + (0.647 − 0.114i)7-s + (0.688 − 0.397i)8-s + (0.0225 + 0.999i)9-s + (0.884 + 0.321i)11-s + (−0.246 + 0.176i)12-s + (0.337 − 0.402i)13-s + (0.575 + 0.482i)14-s + (1.13 + 0.414i)16-s + (−1.34 − 0.778i)17-s + (−0.857 + 0.753i)18-s + (−0.558 − 0.967i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.109 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.29922 + 2.05963i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.29922 + 2.05963i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.23 - 1.21i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (-1.03 - 1.23i)T + (-0.347 + 1.96i)T^{2} \) |
| 7 | \( 1 + (-1.71 + 0.302i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.93 - 1.06i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.21 + 1.44i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (5.56 + 3.21i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.43 + 4.21i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.73 + 0.481i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.94 - 4.98i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.687 - 3.89i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.704 + 0.406i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.80 - 4.86i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.901 - 2.47i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (7.03 - 1.24i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 1.86iT - 53T^{2} \) |
| 59 | \( 1 + (0.971 - 0.353i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.78 + 10.1i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.76 + 10.4i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.06 + 7.03i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (8.53 - 4.92i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-11.3 + 9.56i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (5.92 + 7.06i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-6.28 - 10.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.39 + 14.8i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−10.84187428811258693405858759331, −9.636686647195102370198443578893, −8.939723581205023805014841435824, −7.980759936550222676419818468592, −7.09682729025215052426545002785, −6.24995230709484139308407568120, −4.89882560258140593896985780450, −4.57395793662979583730055541035, −3.45509220535545195524857332688, −1.87198864730275629675092479639,
1.64237742594467920450898533647, 2.27634193278807308696379066844, 3.83438007716546518105027965209, 4.12236282268814924399541828136, 5.77087312630096008690946615762, 6.70493648518675406553455365033, 7.88352272667074443441799877890, 8.460290181229207125376246120096, 9.381235023070166687969730680124, 10.59222256604962832598761198935
