| L(s) = 1 | + (−1.22 + 0.707i)2-s + (−2.92 − 0.682i)3-s + (0.999 − 1.73i)4-s + (2.76 + 1.59i)5-s + (4.06 − 1.23i)6-s + (−1.32 − 2.29i)7-s + 2.82i·8-s + (8.06 + 3.98i)9-s − 4.52·10-s + (−16.3 + 9.42i)11-s + (−4.10 + 4.37i)12-s + (−5.18 + 8.98i)13-s + (3.24 + 1.87i)14-s + (−6.99 − 6.55i)15-s + (−2.00 − 3.46i)16-s + 24.4i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.973 − 0.227i)3-s + (0.249 − 0.433i)4-s + (0.553 + 0.319i)5-s + (0.676 − 0.205i)6-s + (−0.188 − 0.327i)7-s + 0.353i·8-s + (0.896 + 0.442i)9-s − 0.452·10-s + (−1.48 + 0.857i)11-s + (−0.341 + 0.364i)12-s + (−0.398 + 0.690i)13-s + (0.231 + 0.133i)14-s + (−0.466 − 0.437i)15-s + (−0.125 − 0.216i)16-s + 1.43i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.173676 + 0.406332i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.173676 + 0.406332i\) |
| \(L(2)\) |
|
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 | \( 1 + (2.92 + 0.682i)T \) |
| 7 | \( 1 + (1.32 + 2.29i)T \) |
| good | 5 | \( 1 + (-2.76 - 1.59i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (16.3 - 9.42i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (5.18 - 8.98i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 24.4iT - 289T^{2} \) |
| 19 | \( 1 + 26.5T + 361T^{2} \) |
| 23 | \( 1 + (-35.3 - 20.4i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (18.8 - 10.8i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-1.37 + 2.38i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 45.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + (12.8 + 7.41i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-14.4 - 25.0i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (25.7 - 14.8i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 18.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (46.4 + 26.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-13.9 - 24.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-46.2 + 80.1i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 81.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 81.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-42.3 - 73.2i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (99.9 - 57.6i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 13.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-55.7 - 96.6i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−13.21931541837289844044001296061, −12.64610934976458642840207560682, −11.07386267194619380222427929070, −10.45268801206753785698912298615, −9.576379809362778708495779106346, −7.925062828368973750517924107366, −6.91877100794942436196424368925, −5.97083528627995090669453534272, −4.68637152333313504743881601789, −1.98733099480683358123276633236,
0.39640215372810472219819680234, 2.72407145255959531481661741975, 4.92177738323170375429478146918, 5.89559127252259420241537810460, 7.32591938171408503557644617288, 8.710641873943782124045037548752, 9.805225761279705292080093798550, 10.68004899516021466360417807721, 11.45429938203616842819846420913, 12.81960534836990120134059159271
