L(s) = 1 | + (2.68 + 1.34i)3-s + (−3.34 + 1.93i)5-s + (5.38 + 7.20i)9-s + (−12.2 − 7.05i)11-s − 20.2·13-s + (−11.5 + 0.685i)15-s + (−18.9 − 10.9i)17-s + (3.46 + 6.00i)19-s + (6.69 − 3.86i)23-s + (−5.02 + 8.69i)25-s + (4.76 + 26.5i)27-s − 37.3i·29-s + (7.64 − 13.2i)31-s + (−23.2 − 35.3i)33-s + (6.06 + 10.5i)37-s + ⋯ |
L(s) = 1 | + (0.894 + 0.447i)3-s + (−0.669 + 0.386i)5-s + (0.598 + 0.801i)9-s + (−1.11 − 0.640i)11-s − 1.55·13-s + (−0.772 + 0.0456i)15-s + (−1.11 − 0.642i)17-s + (0.182 + 0.316i)19-s + (0.291 − 0.168i)23-s + (−0.200 + 0.347i)25-s + (0.176 + 0.984i)27-s − 1.28i·29-s + (0.246 − 0.427i)31-s + (−0.705 − 1.07i)33-s + (0.163 + 0.283i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1986921184\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1986921184\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.68 - 1.34i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (3.34 - 1.93i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (12.2 + 7.05i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 20.2T + 169T^{2} \) |
| 17 | \( 1 + (18.9 + 10.9i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-3.46 - 6.00i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-6.69 + 3.86i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 37.3iT - 841T^{2} \) |
| 31 | \( 1 + (-7.64 + 13.2i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-6.06 - 10.5i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 42.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 16.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + (62.2 - 35.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (87.8 + 50.7i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (22.2 + 12.8i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-39.0 - 67.6i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-61.5 + 106. i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 34.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (50.2 - 87.0i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-21.2 - 36.8i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 82.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-36.6 + 21.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 48.3T + 9.40e3T^{2} \) |
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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
−10.96682422255256465056686356474, −9.919947460620834833137274392304, −9.377729877797423299473798818534, −8.035330165096605850313147417916, −7.80461949509953185719738203537, −6.70001588574628375330280178783, −5.16017607041899197874806394978, −4.37097472338566566101195355177, −3.10376687300684622642069907284, −2.38386106660481815365713878108,
0.06057080629647432757729131527, 1.96485812943584707108312952070, 2.96348799859861117011862415796, 4.30708794191141907600020102246, 5.10436166371329437386538331698, 6.75353843670081927406622854923, 7.45045754510834571748192717486, 8.156202251138080096438927097501, 9.006433746315537127634571839879, 9.876606304136565473440889544774