L(s) = 1 | + (−0.0569 + 0.377i)2-s + (0.421 − 1.68i)3-s + (1.77 + 0.546i)4-s + (−0.273 − 3.64i)5-s + (0.610 + 0.254i)6-s + (−2.49 + 0.882i)7-s + (−0.639 + 1.32i)8-s + (−2.64 − 1.41i)9-s + (1.39 + 0.104i)10-s + (−0.795 + 0.312i)11-s + (1.66 − 2.74i)12-s + (3.66 + 2.92i)13-s + (−0.191 − 0.992i)14-s + (−6.24 − 1.07i)15-s + (2.59 + 1.77i)16-s + (3.49 + 3.24i)17-s + ⋯ |
L(s) = 1 | + (−0.0402 + 0.267i)2-s + (0.243 − 0.970i)3-s + (0.885 + 0.273i)4-s + (−0.122 − 1.63i)5-s + (0.249 + 0.104i)6-s + (−0.942 + 0.333i)7-s + (−0.225 + 0.469i)8-s + (−0.881 − 0.471i)9-s + (0.440 + 0.0330i)10-s + (−0.239 + 0.0941i)11-s + (0.480 − 0.792i)12-s + (1.01 + 0.809i)13-s + (−0.0511 − 0.265i)14-s + (−1.61 − 0.277i)15-s + (0.649 + 0.442i)16-s + (0.848 + 0.787i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13731 - 0.542801i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13731 - 0.542801i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.421 + 1.68i)T \) |
| 7 | \( 1 + (2.49 - 0.882i)T \) |
good | 2 | \( 1 + (0.0569 - 0.377i)T + (-1.91 - 0.589i)T^{2} \) |
| 5 | \( 1 + (0.273 + 3.64i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (0.795 - 0.312i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-3.66 - 2.92i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-3.49 - 3.24i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.80 + 1.62i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.30 - 2.48i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (0.0736 + 0.0168i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (4.42 + 2.55i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.80 + 0.864i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (2.08 + 1.00i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (1.81 - 0.876i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (13.3 + 2.01i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (1.59 - 5.16i)T + (-43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.0626 - 0.836i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (4.09 + 13.2i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (2.98 - 5.16i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.77 - 1.31i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (0.552 + 3.66i)T + (-69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-1.04 - 1.80i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.07 - 10.1i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-5.23 + 13.3i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 4.03iT - 97T^{2} \) |
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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
−12.84199102004309744996800313872, −12.16546191318381479814420014099, −11.31924201060143063235473723981, −9.441217642058268305783991079539, −8.573659617827151514808459772011, −7.69252797783458919256706767563, −6.46094299880752397599364658730, −5.53521349012604343105549746366, −3.42365935639681374411034338512, −1.57152884490621553073597106503,
3.03771470330309179278136127544, 3.33998049124595249348866526208, 5.69954450745477086637252056983, 6.71321404546708471728222100270, 7.79926958162876683052474690155, 9.593326627233701231723624064232, 10.37960539818369533122160001998, 10.84324660459464551898896760814, 11.78908762071691061670415070785, 13.35404130882193489894335671077