| L(s) = 1 | + (−1.40 + 0.118i)2-s + (0.540 + 0.841i)3-s + (1.97 − 0.332i)4-s + (−3.96 + 1.81i)5-s + (−0.861 − 1.12i)6-s + (−1.61 + 0.475i)7-s + (−2.74 + 0.701i)8-s + (−0.415 + 0.909i)9-s + (5.37 − 3.01i)10-s + (−0.819 + 0.710i)11-s + (1.34 + 1.47i)12-s + (0.113 − 0.386i)13-s + (2.22 − 0.861i)14-s + (−3.66 − 2.35i)15-s + (3.77 − 1.31i)16-s + (0.783 − 5.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.996 + 0.0834i)2-s + (0.312 + 0.485i)3-s + (0.986 − 0.166i)4-s + (−1.77 + 0.809i)5-s + (−0.351 − 0.457i)6-s + (−0.612 + 0.179i)7-s + (−0.968 + 0.248i)8-s + (−0.138 + 0.303i)9-s + (1.69 − 0.954i)10-s + (−0.247 + 0.214i)11-s + (0.388 + 0.427i)12-s + (0.0314 − 0.107i)13-s + (0.595 − 0.230i)14-s + (−0.946 − 0.608i)15-s + (0.944 − 0.328i)16-s + (0.190 − 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0111 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0111 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.132954 - 0.131474i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.132954 - 0.131474i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.40 - 0.118i)T \) |
| 3 | \( 1 + (-0.540 - 0.841i)T \) |
| 23 | \( 1 + (-3.43 + 3.34i)T \) |
| good | 5 | \( 1 + (3.96 - 1.81i)T + (3.27 - 3.77i)T^{2} \) |
| 7 | \( 1 + (1.61 - 0.475i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (0.819 - 0.710i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.113 + 0.386i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.783 + 5.45i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.673 + 0.0968i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (2.81 + 0.404i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (2.24 + 1.44i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (3.01 + 1.37i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-3.05 - 6.68i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.367 - 0.572i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 2.72T + 47T^{2} \) |
| 53 | \( 1 + (0.523 + 1.78i)T + (-44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-2.35 + 8.03i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (3.07 - 4.78i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (10.4 + 9.06i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-6.46 + 7.46i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (2.20 + 15.3i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (13.5 + 3.99i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (5.70 + 2.60i)T + (54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (14.2 - 9.15i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-6.00 - 13.1i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.57767411229511160615892097838, −9.630152418341845177078779966887, −8.818072236787450497843707652372, −7.80838489020574439671374898082, −7.32721940733570516877964045733, −6.39625359042723913972446553547, −4.79098837195274924838907114451, −3.39960649427194015678666668384, −2.77706615653063401251819254589, −0.15447651223068698617256487470,
1.28931471918931136089829241340, 3.17989951796752491549181502098, 3.97168816263112638370443933328, 5.65374885151571719570738227205, 7.04776086315526360555718558007, 7.52193063876820381117786328910, 8.494967152714270468712662819385, 8.833545258329403789002795829779, 10.04484106697835269235014200365, 11.12203448817801211263629433393
