| L(s) = 1 | + (1.34 − 0.451i)2-s + (1.50 − 0.853i)3-s + (1.59 − 1.21i)4-s + (0.248 − 0.683i)5-s + (1.63 − 1.82i)6-s + (−0.984 − 0.173i)7-s + (1.58 − 2.34i)8-s + (1.54 − 2.57i)9-s + (0.0248 − 1.02i)10-s + (0.984 − 0.358i)11-s + (1.36 − 3.18i)12-s + (−2.69 + 2.26i)13-s + (−1.39 + 0.211i)14-s + (−0.208 − 1.24i)15-s + (1.07 − 3.85i)16-s + (−3.34 + 1.93i)17-s + ⋯ |
| L(s) = 1 | + (0.947 − 0.319i)2-s + (0.870 − 0.492i)3-s + (0.796 − 0.605i)4-s + (0.111 − 0.305i)5-s + (0.667 − 0.744i)6-s + (−0.372 − 0.0656i)7-s + (0.561 − 0.827i)8-s + (0.514 − 0.857i)9-s + (0.00785 − 0.325i)10-s + (0.296 − 0.108i)11-s + (0.394 − 0.918i)12-s + (−0.748 + 0.628i)13-s + (−0.373 + 0.0566i)14-s + (−0.0537 − 0.320i)15-s + (0.267 − 0.963i)16-s + (−0.810 + 0.468i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.75125 - 2.20961i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.75125 - 2.20961i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.34 + 0.451i)T \) |
| 3 | \( 1 + (-1.50 + 0.853i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| good | 5 | \( 1 + (-0.248 + 0.683i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-0.984 + 0.358i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (2.69 - 2.26i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.34 - 1.93i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.50 - 1.44i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0714 + 0.405i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (2.28 - 2.72i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (3.49 - 0.616i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.18 - 5.51i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.89 - 7.02i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.35 + 6.47i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.21 + 6.88i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 3.48iT - 53T^{2} \) |
| 59 | \( 1 + (-10.3 - 3.76i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.816 - 4.63i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (5.92 + 7.06i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.261 - 0.452i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.28 - 12.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.06 - 8.41i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-4.99 - 4.18i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (4.22 + 2.44i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.917 - 0.333i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.07742798205481022832304530166, −9.388424110763920993754781033435, −8.510044531357265607336552621401, −7.23267387083050571005202586899, −6.76871449831186024583690914225, −5.65303668576961443616929041616, −4.47703520354448788156919787388, −3.57888462047618651097543450361, −2.52587459548790281817483725894, −1.41828920351484544078424644427,
2.31536946021490696871501136998, 3.03173553957285231764107726559, 4.11672717319649759144261234809, 4.96072299424237841404743119901, 6.02253215138635171688130163429, 7.16316780548141952394969724817, 7.66442926052844379979421700107, 8.841760151732437254920958670973, 9.610959575298379778000311753004, 10.58819580713281052531996098322
