| L(s) = 1 | + (0.729 − 1.26i)2-s + (−1.15 + 0.836i)3-s + (−0.0646 − 0.112i)4-s + (2.41 + 2.68i)5-s + (0.216 + 2.06i)6-s + (−1.66 − 1.85i)7-s + 2.72·8-s + (−0.301 + 0.928i)9-s + (5.15 − 1.09i)10-s + (3.47 − 1.54i)11-s + (0.168 + 0.0748i)12-s + (−3.94 − 1.75i)13-s + (−3.56 + 0.756i)14-s + (−5.02 − 1.06i)15-s + (2.12 − 3.67i)16-s + (−5.79 − 1.23i)17-s + ⋯ |
| L(s) = 1 | + (0.515 − 0.893i)2-s + (−0.664 + 0.482i)3-s + (−0.0323 − 0.0560i)4-s + (1.08 + 1.20i)5-s + (0.0885 + 0.842i)6-s + (−0.630 − 0.700i)7-s + 0.965·8-s + (−0.100 + 0.309i)9-s + (1.63 − 0.346i)10-s + (1.04 − 0.466i)11-s + (0.0485 + 0.0216i)12-s + (−1.09 − 0.486i)13-s + (−0.951 + 0.202i)14-s + (−1.29 − 0.276i)15-s + (0.530 − 0.918i)16-s + (−1.40 − 0.298i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38371 - 0.0697791i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38371 - 0.0697791i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (9.91 - 7.25i)T \) |
| good | 2 | \( 1 + (-0.729 + 1.26i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.15 - 0.836i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (-2.41 - 2.68i)T + (-0.522 + 4.97i)T^{2} \) |
| 7 | \( 1 + (1.66 + 1.85i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-3.47 + 1.54i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (3.94 + 1.75i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (5.79 + 1.23i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 - 4.95T + 19T^{2} \) |
| 23 | \( 1 + (-0.352 + 0.610i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.52 + 1.10i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (6.73 + 1.43i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (-0.970 - 9.22i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (-5.08 + 3.69i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-3.32 + 3.69i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (9.30 + 4.14i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (1.34 + 0.973i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + 6.77T + 59T^{2} \) |
| 61 | \( 1 + (-0.730 + 6.94i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-2.55 - 7.86i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-8.70 + 1.85i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (1.83 - 5.64i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (1.29 - 3.98i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (2.50 + 7.69i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (5.14 - 5.70i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-17.8 - 3.79i)T + (88.6 + 39.4i)T^{2} \) |
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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
−13.08480202052460015607244919931, −11.66819814641116212780091694350, −11.03944954917555661872409895675, −10.24923431068164105603109734000, −9.557866515494088574592587168184, −7.35322820144623292826266534492, −6.43180167500212683393612926339, −5.06856616662758939537966056720, −3.60765214467632179539645128130, −2.38834240990923899105728551912,
1.72179234757467651023687818999, 4.59284055544618501125620727012, 5.62688007189515103802187379059, 6.29733584498998419229193913788, 7.22236202037274973372741697912, 9.204878588729022273205307745824, 9.459910204863949979808237740799, 11.28698571361783436488448567631, 12.50222180796178330329137547494, 12.85848072906517159512715908788
