| 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
−11.01884712851723567075916811925, −10.16044511121878814498265503954, −9.317848342984828053973422508659, −8.581962778166136051628056536227, −7.19497122325615775088896604048, −6.50853257156169787427104934271, −6.22590170261323016782076864170, −4.52106296402836908263917686694, −4.26717423728787691573168185881, −2.92116633003179762032076425289,
0.06032766605751652697915697863, 1.33273726247273310165968557114, 2.62923730599070886491338392548, 4.23539883694832181655921412828, 4.86614988820663132455944482081, 5.82634197361959349839718671451, 6.90608801258837081480945992786, 8.081222225239467831499647982526, 8.815847305299852762121784710647, 9.845623951707724867207353409179
