| L(s) = 1 | + (0.277 − 1.38i)2-s + (−1.26 − 1.18i)3-s + (−1.84 − 0.769i)4-s + (−0.868 − 2.38i)5-s + (−1.99 + 1.42i)6-s + (−0.984 + 0.173i)7-s + (−1.57 + 2.34i)8-s + (0.206 + 2.99i)9-s + (−3.54 + 0.541i)10-s + (3.37 + 1.22i)11-s + (1.42 + 3.15i)12-s + (−4.51 − 3.78i)13-s + (−0.0325 + 1.41i)14-s + (−1.72 + 4.04i)15-s + (2.81 + 2.84i)16-s + (−3.47 − 2.00i)17-s + ⋯ |
| L(s) = 1 | + (0.196 − 0.980i)2-s + (−0.731 − 0.682i)3-s + (−0.922 − 0.384i)4-s + (−0.388 − 1.06i)5-s + (−0.812 + 0.582i)6-s + (−0.372 + 0.0656i)7-s + (−0.558 + 0.829i)8-s + (0.0687 + 0.997i)9-s + (−1.12 + 0.171i)10-s + (1.01 + 0.370i)11-s + (0.412 + 0.911i)12-s + (−1.25 − 1.05i)13-s + (−0.00869 + 0.377i)14-s + (−0.444 + 1.04i)15-s + (0.703 + 0.710i)16-s + (−0.841 − 0.486i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0967041 + 0.0467581i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0967041 + 0.0467581i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.277 + 1.38i)T \) |
| 3 | \( 1 + (1.26 + 1.18i)T \) |
| 7 | \( 1 + (0.984 - 0.173i)T \) |
| good | 5 | \( 1 + (0.868 + 2.38i)T + (-3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-3.37 - 1.22i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (4.51 + 3.78i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.47 + 2.00i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.13 - 2.96i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.655 - 3.71i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.14 - 1.36i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-9.44 - 1.66i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.758 + 1.31i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.88 - 4.63i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.65 + 7.30i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.441 - 2.50i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 7.64iT - 53T^{2} \) |
| 59 | \( 1 + (6.58 - 2.39i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.237 + 1.34i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (7.87 - 9.39i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (6.89 - 11.9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.359 + 0.623i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.79 + 10.4i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (3.06 - 2.56i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (14.0 - 8.13i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (11.5 + 4.20i)T + (74.3 + 62.3i)T^{2} \) |
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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
−9.845623951707724867207353409179, −8.815847305299852762121784710647, −8.081222225239467831499647982526, −6.90608801258837081480945992786, −5.82634197361959349839718671451, −4.86614988820663132455944482081, −4.23539883694832181655921412828, −2.62923730599070886491338392548, −1.33273726247273310165968557114, −0.06032766605751652697915697863,
2.92116633003179762032076425289, 4.26717423728787691573168185881, 4.52106296402836908263917686694, 6.22590170261323016782076864170, 6.50853257156169787427104934271, 7.19497122325615775088896604048, 8.581962778166136051628056536227, 9.317848342984828053973422508659, 10.16044511121878814498265503954, 11.01884712851723567075916811925
