| L(s) = 1 | + (−0.868 + 1.28i)2-s + (−0.112 + 1.72i)3-s + (−0.146 − 0.368i)4-s + (0.104 − 0.224i)5-s + (−2.11 − 1.64i)6-s + (0.721 + 2.59i)7-s + (−2.42 − 0.533i)8-s + (−2.97 − 0.389i)9-s + (0.197 + 0.328i)10-s + (3.37 + 3.19i)11-s + (0.653 − 0.212i)12-s + (0.511 − 4.70i)13-s + (−3.95 − 1.33i)14-s + (0.376 + 0.205i)15-s + (3.36 − 3.18i)16-s + (−5.31 − 1.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.614 + 0.906i)2-s + (−0.0650 + 0.997i)3-s + (−0.0734 − 0.184i)4-s + (0.0465 − 0.100i)5-s + (−0.864 − 0.671i)6-s + (0.272 + 0.981i)7-s + (−0.856 − 0.188i)8-s + (−0.991 − 0.129i)9-s + (0.0625 + 0.103i)10-s + (1.01 + 0.963i)11-s + (0.188 − 0.0613i)12-s + (0.142 − 1.30i)13-s + (−1.05 − 0.356i)14-s + (0.0973 + 0.0529i)15-s + (0.841 − 0.797i)16-s + (−1.29 − 0.358i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0665441 + 0.797285i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0665441 + 0.797285i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.112 - 1.72i)T \) |
| 59 | \( 1 + (-6.24 + 4.47i)T \) |
| good | 2 | \( 1 + (0.868 - 1.28i)T + (-0.740 - 1.85i)T^{2} \) |
| 5 | \( 1 + (-0.104 + 0.224i)T + (-3.23 - 3.81i)T^{2} \) |
| 7 | \( 1 + (-0.721 - 2.59i)T + (-5.99 + 3.60i)T^{2} \) |
| 11 | \( 1 + (-3.37 - 3.19i)T + (0.595 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.511 + 4.70i)T + (-12.6 - 2.79i)T^{2} \) |
| 17 | \( 1 + (5.31 + 1.47i)T + (14.5 + 8.76i)T^{2} \) |
| 19 | \( 1 + (2.81 + 2.14i)T + (5.08 + 18.3i)T^{2} \) |
| 23 | \( 1 + (-3.69 - 6.96i)T + (-12.9 + 19.0i)T^{2} \) |
| 29 | \( 1 + (-3.87 + 2.63i)T + (10.7 - 26.9i)T^{2} \) |
| 31 | \( 1 + (-0.174 - 0.229i)T + (-8.29 + 29.8i)T^{2} \) |
| 37 | \( 1 + (-2.05 - 9.34i)T + (-33.5 + 15.5i)T^{2} \) |
| 41 | \( 1 + (-6.09 - 3.23i)T + (23.0 + 33.9i)T^{2} \) |
| 43 | \( 1 + (1.05 + 1.11i)T + (-2.32 + 42.9i)T^{2} \) |
| 47 | \( 1 + (-0.0458 + 0.0212i)T + (30.4 - 35.8i)T^{2} \) |
| 53 | \( 1 + (1.21 - 2.01i)T + (-24.8 - 46.8i)T^{2} \) |
| 61 | \( 1 + (0.518 + 0.351i)T + (22.5 + 56.6i)T^{2} \) |
| 67 | \( 1 + (1.33 - 6.05i)T + (-60.8 - 28.1i)T^{2} \) |
| 71 | \( 1 + (3.81 + 8.23i)T + (-45.9 + 54.1i)T^{2} \) |
| 73 | \( 1 + (-0.196 + 0.583i)T + (-58.1 - 44.1i)T^{2} \) |
| 79 | \( 1 + (-0.518 + 9.55i)T + (-78.5 - 8.54i)T^{2} \) |
| 83 | \( 1 + (1.54 - 9.40i)T + (-78.6 - 26.5i)T^{2} \) |
| 89 | \( 1 + (7.00 + 10.3i)T + (-32.9 + 82.6i)T^{2} \) |
| 97 | \( 1 + (3.08 + 9.16i)T + (-77.2 + 58.7i)T^{2} \) |
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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
−13.14667805019166543060293701371, −11.97961064484016429735417554595, −11.15772314538089687321907257454, −9.726625625641978177428834232743, −9.050567280027901963787489754932, −8.323385004335407203410788278814, −6.94540515169092458245272189526, −5.81630682759021340395833575134, −4.68331956764531305339793676232, −2.96355137930403788207954184949,
0.953911932597801766938538565495, 2.33792996649785354108358246424, 4.12337763538750377418345521343, 6.26776296896850861999727402305, 6.83404983985114158199113223038, 8.526983999854434316312834391552, 9.012523646262945445267988544037, 10.71172845216481396695557436879, 11.03869045714474494722685066042, 12.02284792980311170624799551658
