| L(s) = 1 | + (−1.59 + 2.24i)2-s + (−0.235 + 0.971i)3-s + (−1.82 − 5.27i)4-s + (−0.128 + 2.70i)5-s + (−1.80 − 2.08i)6-s + (2.64 + 0.0748i)7-s + (9.46 + 2.77i)8-s + (−0.888 − 0.458i)9-s + (−5.86 − 4.61i)10-s + (3.33 + 4.67i)11-s + (5.55 − 0.530i)12-s + (−0.311 + 2.16i)13-s + (−4.39 + 5.81i)14-s + (−2.60 − 0.763i)15-s + (−12.5 + 9.88i)16-s + (−2.11 + 0.407i)17-s + ⋯ |
| L(s) = 1 | + (−1.12 + 1.58i)2-s + (−0.136 + 0.561i)3-s + (−0.912 − 2.63i)4-s + (−0.0576 + 1.21i)5-s + (−0.736 − 0.849i)6-s + (0.999 + 0.0282i)7-s + (3.34 + 0.982i)8-s + (−0.296 − 0.152i)9-s + (−1.85 − 1.45i)10-s + (1.00 + 1.41i)11-s + (1.60 − 0.153i)12-s + (−0.0864 + 0.601i)13-s + (−1.17 + 1.55i)14-s + (−0.671 − 0.197i)15-s + (−3.14 + 2.47i)16-s + (−0.513 + 0.0989i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.191636 - 0.764461i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.191636 - 0.764461i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.235 - 0.971i)T \) |
| 7 | \( 1 + (-2.64 - 0.0748i)T \) |
| 23 | \( 1 + (2.73 + 3.93i)T \) |
| good | 2 | \( 1 + (1.59 - 2.24i)T + (-0.654 - 1.89i)T^{2} \) |
| 5 | \( 1 + (0.128 - 2.70i)T + (-4.97 - 0.475i)T^{2} \) |
| 11 | \( 1 + (-3.33 - 4.67i)T + (-3.59 + 10.3i)T^{2} \) |
| 13 | \( 1 + (0.311 - 2.16i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (2.11 - 0.407i)T + (15.7 - 6.31i)T^{2} \) |
| 19 | \( 1 + (-7.06 - 1.36i)T + (17.6 + 7.06i)T^{2} \) |
| 29 | \( 1 + (-2.79e-5 - 3.22e-5i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (1.97 - 1.88i)T + (1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (4.79 + 2.47i)T + (21.4 + 30.1i)T^{2} \) |
| 41 | \( 1 + (-2.03 - 1.30i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-6.37 + 1.87i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (2.33 + 4.04i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.59 + 1.83i)T + (38.3 + 36.5i)T^{2} \) |
| 59 | \( 1 + (-3.04 - 2.39i)T + (13.9 + 57.3i)T^{2} \) |
| 61 | \( 1 + (2.11 + 8.73i)T + (-54.2 + 27.9i)T^{2} \) |
| 67 | \( 1 + (10.8 + 1.03i)T + (65.7 + 12.6i)T^{2} \) |
| 71 | \( 1 + (3.52 + 7.71i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-1.45 - 4.19i)T + (-57.3 + 45.1i)T^{2} \) |
| 79 | \( 1 + (4.13 - 1.65i)T + (57.1 - 54.5i)T^{2} \) |
| 83 | \( 1 + (-2.47 + 1.58i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (1.13 + 1.07i)T + (4.23 + 88.8i)T^{2} \) |
| 97 | \( 1 + (0.371 + 0.238i)T + (40.2 + 88.2i)T^{2} \) |
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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
−11.03632344250699345561967208515, −10.29848876922858024327897691592, −9.545968858274542262320406330322, −8.826846626639652258180069557536, −7.64546672862839406706235489383, −7.07663743783954464947656259205, −6.31315630782510336201231534329, −5.13641169337469324697595098811, −4.22428039444362360897457180647, −1.79034189748461935661818064591,
0.823280560443203560251121541632, 1.51930212279542031141994711691, 3.10287257004989332785722092095, 4.30360701444715010460914011224, 5.51582411722969248041426208177, 7.43179880218217038591996353987, 8.141705247773470324198191185190, 8.860848564702519745214086982500, 9.392438182839365643611212140518, 10.67119573545529369404881771971
