| 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
−13.35404130882193489894335671077, −11.78908762071691061670415070785, −10.84324660459464551898896760814, −10.37960539818369533122160001998, −9.593326627233701231723624064232, −7.79926958162876683052474690155, −6.71321404546708471728222100270, −5.69954450745477086637252056983, −3.33998049124595249348866526208, −3.03771470330309179278136127544,
1.57152884490621553073597106503, 3.42365935639681374411034338512, 5.53521349012604343105549746366, 6.46094299880752397599364658730, 7.69252797783458919256706767563, 8.573659617827151514808459772011, 9.441217642058268305783991079539, 11.31924201060143063235473723981, 12.16546191318381479814420014099, 12.84199102004309744996800313872
