| L(s) = 1 | + (−0.831 + 1.82i)2-s + (−0.959 − 0.281i)3-s + (−1.31 − 1.51i)4-s + (−0.841 + 0.540i)5-s + (1.31 − 1.51i)6-s + (0.450 − 3.13i)7-s + (0.0155 − 0.00456i)8-s + (0.841 + 0.540i)9-s + (−0.284 − 1.98i)10-s + (−0.0600 − 0.131i)11-s + (0.834 + 1.82i)12-s + (−0.435 − 3.03i)13-s + (5.33 + 3.42i)14-s + (0.959 − 0.281i)15-s + (0.566 − 3.94i)16-s + (4.78 − 5.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.588 + 1.28i)2-s + (−0.553 − 0.162i)3-s + (−0.657 − 0.758i)4-s + (−0.376 + 0.241i)5-s + (0.535 − 0.617i)6-s + (0.170 − 1.18i)7-s + (0.00550 − 0.00161i)8-s + (0.280 + 0.180i)9-s + (−0.0900 − 0.626i)10-s + (−0.0180 − 0.0396i)11-s + (0.240 + 0.527i)12-s + (−0.120 − 0.840i)13-s + (1.42 + 0.916i)14-s + (0.247 − 0.0727i)15-s + (0.141 − 0.985i)16-s + (1.16 − 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.644978 + 0.0221847i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.644978 + 0.0221847i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.959 + 0.281i)T \) |
| 5 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (2.97 - 3.75i)T \) |
| good | 2 | \( 1 + (0.831 - 1.82i)T + (-1.30 - 1.51i)T^{2} \) |
| 7 | \( 1 + (-0.450 + 3.13i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (0.0600 + 0.131i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.435 + 3.03i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-4.78 + 5.52i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-2.03 - 2.35i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-1.17 + 1.35i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-9.74 + 2.86i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (8.11 + 5.21i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (3.39 - 2.18i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (3.77 + 1.10i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 5.23T + 47T^{2} \) |
| 53 | \( 1 + (-1.58 + 10.9i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.377 + 2.62i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (2.51 - 0.738i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-0.210 + 0.459i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-4.77 + 10.4i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (3.60 + 4.16i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.0815 - 0.566i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (2.83 + 1.82i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (15.3 + 4.51i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-6.32 + 4.06i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.56708063313878488154793706710, −10.27849039326091050583511409414, −9.764943988972925394611581823544, −8.222513835173264401177235570031, −7.57211662347674209327626129168, −7.03039974574011440094375905977, −5.87686032256593171558761285103, −4.94971923521571225790063110318, −3.37780078645797025174171810575, −0.64022111125903541058789085899,
1.45225923587860137867925887361, 2.86083682592508786635945254894, 4.21021302695531834968262472156, 5.49922703480781374947466979403, 6.59156837388437966744455453673, 8.270536505057277018306490990061, 8.850562291259489956964529008709, 9.925800683538100050572634876019, 10.55633406792403248129811823183, 11.67619184675221277257519105719
