| L(s) = 1 | + (1.71 − 0.674i)2-s + (1.73 + 0.0727i)3-s + (1.03 − 0.960i)4-s + (−2.08 + 1.42i)5-s + (3.02 − 1.04i)6-s + (−1.82 − 1.92i)7-s + (−0.471 + 0.979i)8-s + (2.98 + 0.251i)9-s + (−2.63 + 3.85i)10-s + (0.349 − 2.32i)11-s + (1.86 − 1.58i)12-s + (−2.55 − 2.03i)13-s + (−4.42 − 2.07i)14-s + (−3.71 + 2.31i)15-s + (−0.360 + 4.81i)16-s + (−5.16 + 1.59i)17-s + ⋯ |
| L(s) = 1 | + (1.21 − 0.477i)2-s + (0.999 + 0.0420i)3-s + (0.517 − 0.480i)4-s + (−0.934 + 0.636i)5-s + (1.23 − 0.425i)6-s + (−0.688 − 0.725i)7-s + (−0.166 + 0.346i)8-s + (0.996 + 0.0839i)9-s + (−0.831 + 1.22i)10-s + (0.105 − 0.699i)11-s + (0.537 − 0.457i)12-s + (−0.707 − 0.564i)13-s + (−1.18 − 0.554i)14-s + (−0.960 + 0.597i)15-s + (−0.0902 + 1.20i)16-s + (−1.25 + 0.386i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.05726 - 0.388376i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.05726 - 0.388376i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.73 - 0.0727i)T \) |
| 7 | \( 1 + (1.82 + 1.92i)T \) |
| good | 2 | \( 1 + (-1.71 + 0.674i)T + (1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (2.08 - 1.42i)T + (1.82 - 4.65i)T^{2} \) |
| 11 | \( 1 + (-0.349 + 2.32i)T + (-10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (2.55 + 2.03i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (5.16 - 1.59i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-5.23 - 3.02i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.00 + 6.51i)T + (-19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-8.11 - 1.85i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (3.37 - 1.94i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.70 + 1.58i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (-4.73 - 2.28i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (0.477 - 0.229i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (0.0175 + 0.0447i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (5.77 + 6.22i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (3.06 + 2.08i)T + (21.5 + 54.9i)T^{2} \) |
| 61 | \( 1 + (1.60 - 1.72i)T + (-4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (0.116 + 0.201i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.49 + 0.341i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (6.13 + 2.40i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + (-1.44 + 2.50i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.47 + 1.85i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-4.54 + 0.685i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + 14.2iT - 97T^{2} \) |
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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
−13.04482986402647623083928366005, −12.34170318300705118739984507427, −11.10034732656832263096845234781, −10.25754055189045276536641483581, −8.752061815863368895339577200652, −7.64557523574766539357731551378, −6.52912193869039188289555352508, −4.62821167056862731815878943174, −3.56462124943731701607830014661, −2.87684942792351368232499036458,
2.83369635756783983085637388392, 4.15736114244537652887989930068, 5.01287374612829733778879328702, 6.73785266640565704131238602487, 7.56339403880194125560690650758, 9.069697859802229831196214069547, 9.583229789648503752848378320428, 11.76116695908063700539224848287, 12.38914610696951349626457739980, 13.26787008439976644857949954134
