| L(s) = 1 | + (−0.231 − 1.39i)2-s + (1.44 + 0.598i)3-s + (−1.89 + 0.645i)4-s + (2.07 − 1.05i)5-s + (0.500 − 2.15i)6-s + (0.213 + 0.348i)7-s + (1.33 + 2.49i)8-s + (−0.392 − 0.392i)9-s + (−1.95 − 2.65i)10-s + (1.39 + 0.109i)11-s + (−3.12 − 0.200i)12-s + (−0.0467 − 0.194i)13-s + (0.436 − 0.378i)14-s + (3.63 − 0.286i)15-s + (3.16 − 2.44i)16-s + (1.14 − 0.982i)17-s + ⋯ |
| L(s) = 1 | + (−0.163 − 0.986i)2-s + (0.834 + 0.345i)3-s + (−0.946 + 0.322i)4-s + (0.929 − 0.473i)5-s + (0.204 − 0.879i)6-s + (0.0807 + 0.131i)7-s + (0.473 + 0.880i)8-s + (−0.130 − 0.130i)9-s + (−0.619 − 0.839i)10-s + (0.419 + 0.0330i)11-s + (−0.900 − 0.0577i)12-s + (−0.0129 − 0.0540i)13-s + (0.116 − 0.101i)14-s + (0.939 − 0.0739i)15-s + (0.791 − 0.610i)16-s + (0.278 − 0.238i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.504 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18063 - 0.677167i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18063 - 0.677167i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.231 + 1.39i)T \) |
| 41 | \( 1 + (3.12 - 5.58i)T \) |
| good | 3 | \( 1 + (-1.44 - 0.598i)T + (2.12 + 2.12i)T^{2} \) |
| 5 | \( 1 + (-2.07 + 1.05i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-0.213 - 0.348i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (-1.39 - 0.109i)T + (10.8 + 1.72i)T^{2} \) |
| 13 | \( 1 + (0.0467 + 0.194i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (-1.14 + 0.982i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (5.19 + 1.24i)T + (16.9 + 8.62i)T^{2} \) |
| 23 | \( 1 + (7.30 - 5.31i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-5.29 - 4.52i)T + (4.53 + 28.6i)T^{2} \) |
| 31 | \( 1 + (-0.798 - 2.45i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.0154 - 0.0474i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (-6.26 - 0.992i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (2.42 - 3.96i)T + (-21.3 - 41.8i)T^{2} \) |
| 53 | \( 1 + (3.80 - 4.45i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (1.05 + 1.44i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (9.16 - 1.45i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (0.256 + 3.26i)T + (-66.1 + 10.4i)T^{2} \) |
| 71 | \( 1 + (-1.06 + 13.5i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (-5.44 + 5.44i)T - 73iT^{2} \) |
| 79 | \( 1 + (-1.94 + 4.70i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 2.69iT - 83T^{2} \) |
| 89 | \( 1 + (2.20 - 1.35i)T + (40.4 - 79.2i)T^{2} \) |
| 97 | \( 1 + (1.30 + 16.5i)T + (-95.8 + 15.1i)T^{2} \) |
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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
−12.63078537044495713496942069189, −11.75458457815593168081343958615, −10.45479365513188957389985552302, −9.545924466680398504622122801930, −8.969279460039173704890430887157, −8.019101679219042768241520911719, −6.01093347251534717984493024169, −4.59251714546402513629570570818, −3.25305311316982373247469141821, −1.85294706315225326986715105480,
2.22112475962196717158352544404, 4.15283658165550787387077704298, 5.85042802967270094742008137840, 6.62795855874276254620116216533, 7.945842323902683407774947526782, 8.624820888051289357741220258415, 9.789863596024171501248431508634, 10.58783460858552996190529861109, 12.38857056768727898079493911598, 13.49458603540345820861671951282
