L(s) = 1 | + (1.22 + 0.707i)2-s + (0.999 + 1.73i)4-s + (−7.34 + 4.24i)5-s + (−2.5 + 4.33i)7-s + 2.82i·8-s − 12·10-s + (−7.34 − 4.24i)11-s + (0.5 + 0.866i)13-s + (−6.12 + 3.53i)14-s + (−2.00 + 3.46i)16-s + 25.4i·17-s + 29·19-s + (−14.6 − 8.48i)20-s + (−6 − 10.3i)22-s + (−7.34 + 4.24i)23-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.46 + 0.848i)5-s + (−0.357 + 0.618i)7-s + 0.353i·8-s − 1.20·10-s + (−0.668 − 0.385i)11-s + (0.0384 + 0.0666i)13-s + (−0.437 + 0.252i)14-s + (−0.125 + 0.216i)16-s + 1.49i·17-s + 1.52·19-s + (−0.734 − 0.424i)20-s + (−0.272 − 0.472i)22-s + (−0.319 + 0.184i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.410223 + 1.12707i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.410223 + 1.12707i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (7.34 - 4.24i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (2.5 - 4.33i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (7.34 + 4.24i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 25.4iT - 289T^{2} \) |
| 19 | \( 1 - 29T + 361T^{2} \) |
| 23 | \( 1 + (7.34 - 4.24i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-14.6 - 8.48i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-5 - 8.66i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 25T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-14.6 + 8.48i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (7 - 12.1i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (7.34 + 4.24i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 50.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-80.8 + 46.6i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (11.5 - 19.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-9.5 - 16.4i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 101. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 97T + 5.32e3T^{2} \) |
| 79 | \( 1 + (38.5 - 66.6i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-102. - 59.3i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 76.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-24.5 + 42.4i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−12.92564581734311803271830316203, −12.03183123881232704707156331844, −11.29422460576236159880940199885, −10.24685903288461215783455465506, −8.524414648127735386752547788684, −7.71953122955539998898266217951, −6.68508586845698356336206502199, −5.45941698003918718007645857599, −3.88292612436839142850836556864, −2.96395057147542043121437165680,
0.61990618642850739388196062149, 3.14135425398315323102892716856, 4.34550843520151380277236248954, 5.24551323020402501043498642311, 7.12796704966421837072672758211, 7.85182493129889192832604739590, 9.269490158799491216141707582417, 10.39900904023398192801440473172, 11.67166665400585966522124379973, 12.06638412745222105613493787503