| L(s) = 1 | + (1.03 + 2.59i)3-s + (−0.461 + 1.90i)5-s + (−1.73 − 2.00i)7-s + (−3.49 + 3.32i)9-s + (−3.36 − 0.160i)11-s + (−3.59 − 3.11i)13-s + (−5.41 + 0.778i)15-s + (−3.06 + 4.30i)17-s + (4.83 + 6.79i)19-s + (3.39 − 6.57i)21-s + (−3.78 − 2.93i)23-s + (1.04 + 0.537i)25-s + (−4.63 − 2.11i)27-s + (−2.00 − 4.39i)29-s + (4.21 + 5.35i)31-s + ⋯ |
| L(s) = 1 | + (0.600 + 1.49i)3-s + (−0.206 + 0.850i)5-s + (−0.654 − 0.756i)7-s + (−1.16 + 1.10i)9-s + (−1.01 − 0.0482i)11-s + (−0.997 − 0.864i)13-s + (−1.39 + 0.201i)15-s + (−0.743 + 1.04i)17-s + (1.10 + 1.55i)19-s + (0.740 − 1.43i)21-s + (−0.790 − 0.612i)23-s + (0.208 + 0.107i)25-s + (−0.892 − 0.407i)27-s + (−0.372 − 0.815i)29-s + (0.756 + 0.962i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0198058 - 0.981091i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0198058 - 0.981091i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.73 + 2.00i)T \) |
| 23 | \( 1 + (3.78 + 2.93i)T \) |
| good | 3 | \( 1 + (-1.03 - 2.59i)T + (-2.17 + 2.07i)T^{2} \) |
| 5 | \( 1 + (0.461 - 1.90i)T + (-4.44 - 2.29i)T^{2} \) |
| 11 | \( 1 + (3.36 + 0.160i)T + (10.9 + 1.04i)T^{2} \) |
| 13 | \( 1 + (3.59 + 3.11i)T + (1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.06 - 4.30i)T + (-5.56 - 16.0i)T^{2} \) |
| 19 | \( 1 + (-4.83 - 6.79i)T + (-6.21 + 17.9i)T^{2} \) |
| 29 | \( 1 + (2.00 + 4.39i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-4.21 - 5.35i)T + (-7.30 + 30.1i)T^{2} \) |
| 37 | \( 1 + (-6.42 - 6.73i)T + (-1.76 + 36.9i)T^{2} \) |
| 41 | \( 1 + (-1.23 - 4.20i)T + (-34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (4.33 + 0.622i)T + (41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (6.07 + 3.50i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.79 - 1.65i)T + (41.6 + 32.7i)T^{2} \) |
| 59 | \( 1 + (-1.09 + 5.65i)T + (-54.7 - 21.9i)T^{2} \) |
| 61 | \( 1 + (-12.9 - 5.17i)T + (44.1 + 42.0i)T^{2} \) |
| 67 | \( 1 + (-2.62 + 5.08i)T + (-38.8 - 54.5i)T^{2} \) |
| 71 | \( 1 + (6.50 - 4.17i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (0.356 + 3.73i)T + (-71.6 + 13.8i)T^{2} \) |
| 79 | \( 1 + (9.61 - 3.32i)T + (62.0 - 48.8i)T^{2} \) |
| 83 | \( 1 + (7.51 + 2.20i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-13.1 - 10.3i)T + (20.9 + 86.4i)T^{2} \) |
| 97 | \( 1 + (-2.13 + 0.627i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.38331670205630120215697915311, −10.26698899378413831363282396338, −9.734665062280311970245649805392, −8.288040582781838367664970885573, −7.77979041870442978242135797419, −6.54544647027786730650500923860, −5.36646440821306313130818224789, −4.29424897165275136990886728071, −3.40544202282051968220945195033, −2.70952683709989217748871662325,
0.46046691611630171461004492399, 2.21413412796748308643962264050, 2.84019831818255968150177041879, 4.68358981310122312341100893489, 5.62492642098360745717783472875, 6.92025458263498847711046691240, 7.35440282124129847410599496548, 8.352929312990586636813204849811, 9.147279231986319323294096187251, 9.686305858215717747353522324238
