L(s) = 1 | + (0.900 − 0.433i)2-s + (0.403 + 0.194i)3-s + (0.623 − 0.781i)4-s + (−0.222 + 0.974i)5-s + 0.447·6-s + 2.78·7-s + (0.222 − 0.974i)8-s + (−1.74 − 2.18i)9-s + (0.222 + 0.974i)10-s + (3.02 + 3.79i)11-s + (0.403 − 0.194i)12-s + (0.482 − 2.11i)13-s + (2.50 − 1.20i)14-s + (−0.278 + 0.349i)15-s + (−0.222 − 0.974i)16-s + (0.129 + 0.568i)17-s + ⋯ |
L(s) = 1 | + (0.637 − 0.306i)2-s + (0.232 + 0.112i)3-s + (0.311 − 0.390i)4-s + (−0.0995 + 0.436i)5-s + 0.182·6-s + 1.05·7-s + (0.0786 − 0.344i)8-s + (−0.581 − 0.729i)9-s + (0.0703 + 0.308i)10-s + (0.912 + 1.14i)11-s + (0.116 − 0.0560i)12-s + (0.133 − 0.585i)13-s + (0.669 − 0.322i)14-s + (−0.0720 + 0.0903i)15-s + (−0.0556 − 0.243i)16-s + (0.0314 + 0.137i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.26679 - 0.326837i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.26679 - 0.326837i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.900 + 0.433i)T \) |
| 5 | \( 1 + (0.222 - 0.974i)T \) |
| 43 | \( 1 + (3.73 + 5.39i)T \) |
good | 3 | \( 1 + (-0.403 - 0.194i)T + (1.87 + 2.34i)T^{2} \) |
| 7 | \( 1 - 2.78T + 7T^{2} \) |
| 11 | \( 1 + (-3.02 - 3.79i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.482 + 2.11i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.129 - 0.568i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 + (-0.315 + 0.395i)T + (-4.22 - 18.5i)T^{2} \) |
| 23 | \( 1 + (-1.20 - 1.51i)T + (-5.11 + 22.4i)T^{2} \) |
| 29 | \( 1 + (-2.41 + 1.16i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (2.05 - 0.991i)T + (19.3 - 24.2i)T^{2} \) |
| 37 | \( 1 + 4.50T + 37T^{2} \) |
| 41 | \( 1 + (3.01 - 1.44i)T + (25.5 - 32.0i)T^{2} \) |
| 47 | \( 1 + (3.13 - 3.92i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (-1.64 - 7.22i)T + (-47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 + (2.67 + 11.7i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (7.37 + 3.55i)T + (38.0 + 47.6i)T^{2} \) |
| 67 | \( 1 + (4.32 - 5.41i)T + (-14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (0.495 - 0.620i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (2.26 - 9.93i)T + (-65.7 - 31.6i)T^{2} \) |
| 79 | \( 1 + 8.10T + 79T^{2} \) |
| 83 | \( 1 + (-2.29 - 1.10i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (1.65 + 0.798i)T + (55.4 + 69.5i)T^{2} \) |
| 97 | \( 1 + (-7.43 - 9.32i)T + (-21.5 + 94.5i)T^{2} \) |
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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
−11.33242096108068296819016136517, −10.36322303831040276973595974761, −9.422717977874732828074644209352, −8.404764848998439009700007941712, −7.30112310817344580811415038128, −6.35473866466512059475819110965, −5.18557867761859552732447920578, −4.15483247743974702528498944621, −3.10012455052270574576393611690, −1.65720463114967979849387776673,
1.68636325223092348657991607177, 3.25685016337722120482121162316, 4.49965128838853108542861981235, 5.34517527615400048229222540266, 6.38226845894829702987707368392, 7.57476668366788036860235370979, 8.474247125813890444494210583132, 8.945690991963120133432493858172, 10.59815108828133884583740802097, 11.50116506511089428729710069761