| L(s) = 1 | − 1.41i·2-s + (2.25 − 1.98i)3-s − 2.00·4-s + (2.56 − 1.48i)5-s + (−2.80 − 3.18i)6-s + (6.84 + 1.47i)7-s + 2.82i·8-s + (1.15 − 8.92i)9-s + (−2.09 − 3.62i)10-s + (−9.06 − 5.23i)11-s + (−4.50 + 3.96i)12-s + (−1.61 + 2.79i)13-s + (2.09 − 9.67i)14-s + (2.84 − 8.41i)15-s + 4.00·16-s + (−4.26 + 2.46i)17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (0.751 − 0.660i)3-s − 0.500·4-s + (0.512 − 0.296i)5-s + (−0.466 − 0.531i)6-s + (0.977 + 0.211i)7-s + 0.353i·8-s + (0.128 − 0.991i)9-s + (−0.209 − 0.362i)10-s + (−0.824 − 0.475i)11-s + (−0.375 + 0.330i)12-s + (−0.124 + 0.214i)13-s + (0.149 − 0.691i)14-s + (0.189 − 0.561i)15-s + 0.250·16-s + (−0.251 + 0.144i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.22960 - 1.36372i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22960 - 1.36372i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (-2.25 + 1.98i)T \) |
| 7 | \( 1 + (-6.84 - 1.47i)T \) |
| good | 5 | \( 1 + (-2.56 + 1.48i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (9.06 + 5.23i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (1.61 - 2.79i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (4.26 - 2.46i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (10.3 - 17.8i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-12.7 + 7.38i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-26.9 + 15.5i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 - 26.4T + 961T^{2} \) |
| 37 | \( 1 + (25.0 - 43.3i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-42.6 - 24.6i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-26.9 - 46.6i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + 19.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (86.9 - 50.2i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + 27.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 40.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 76.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 2.51iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (69.1 + 119. i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + 32.6T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-7.18 + 4.14i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (107. + 61.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-44.1 - 76.4i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.90247971785288523070148011036, −11.99021221155204069706798602555, −10.86092481709750262662538795530, −9.666247378180425060666591991307, −8.538390117884773339439187066938, −7.86831948110257127020827558720, −6.09859744944396689977534408357, −4.60552096637533123591311005803, −2.79625494551732291526527245330, −1.49582443396562702126147886024,
2.47786353182935790252666397280, 4.39885907979023841072225434454, 5.33621366641427666236059637813, 7.06962297079725309204054792774, 8.078610259051600255602907464323, 9.038384418221123495594821751836, 10.19778225435137571903470381819, 10.97136467216752193629585667733, 12.74874763097081180906228206447, 13.84960935781301621467113696136
