| L(s) = 1 | + (0.827 − 1.14i)2-s + (1.29 − 1.15i)3-s + (−0.631 − 1.89i)4-s + (0.639 − 1.75i)5-s + (−0.257 − 2.43i)6-s + (0.984 + 0.173i)7-s + (−2.69 − 0.845i)8-s + (0.330 − 2.98i)9-s + (−1.48 − 2.18i)10-s + (0.0740 − 0.0269i)11-s + (−3.00 − 1.71i)12-s + (−0.167 + 0.140i)13-s + (1.01 − 0.985i)14-s + (−1.20 − 3.00i)15-s + (−3.20 + 2.39i)16-s + (−3.21 + 1.85i)17-s + ⋯ |
| L(s) = 1 | + (0.584 − 0.811i)2-s + (0.745 − 0.667i)3-s + (−0.315 − 0.948i)4-s + (0.285 − 0.785i)5-s + (−0.105 − 0.994i)6-s + (0.372 + 0.0656i)7-s + (−0.954 − 0.298i)8-s + (0.110 − 0.993i)9-s + (−0.469 − 0.691i)10-s + (0.0223 − 0.00812i)11-s + (−0.868 − 0.496i)12-s + (−0.0465 + 0.0390i)13-s + (0.270 − 0.263i)14-s + (−0.310 − 0.775i)15-s + (−0.800 + 0.599i)16-s + (−0.780 + 0.450i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.698726 - 2.55569i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.698726 - 2.55569i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.827 + 1.14i)T \) |
| 3 | \( 1 + (-1.29 + 1.15i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
| good | 5 | \( 1 + (-0.639 + 1.75i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-0.0740 + 0.0269i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.167 - 0.140i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.21 - 1.85i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.43 - 1.98i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.995 - 5.64i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.80 + 2.14i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-6.34 + 1.11i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (4.32 + 7.48i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.30 + 1.56i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.501 - 1.37i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.871 - 4.94i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 2.46iT - 53T^{2} \) |
| 59 | \( 1 + (6.52 + 2.37i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.60 - 9.10i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.34 - 3.99i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.37 - 2.38i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.15 + 12.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.27 - 1.52i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.05 + 5.92i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-7.71 - 4.45i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.39 - 0.870i)T + (74.3 - 62.3i)T^{2} \) |
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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
−9.874421791460959112920679739803, −9.184299941875148787224398834654, −8.508968379008103900775998446288, −7.46314014715820698091518396962, −6.28793340018222185051980835407, −5.37008323512951865113141708604, −4.33846944980047989006887841326, −3.27793526989965696389564004907, −2.06681720102035411175575946045, −1.13322511988778115444692839410,
2.51995926897528845402532188150, 3.24443917011733927804157431133, 4.52297790467760048636174413308, 5.08502436865319768465768962877, 6.47974482110103585312077606431, 7.09078206486779370073098368336, 8.164457513399767164551906493151, 8.767637246820180320303511719357, 9.741788570783321182503252073748, 10.61337773901639994262750863109
