| L(s) = 1 | + (−1.22 − 0.705i)2-s + (1.00 + 1.72i)4-s − 3.64i·5-s + (−0.0143 − 2.82i)8-s + (−2.57 + 4.47i)10-s + 6.07i·11-s − 0.483i·13-s + (−1.97 + 3.47i)16-s − 2.55i·17-s − 1.21·19-s + (6.30 − 3.66i)20-s + (4.27 − 7.44i)22-s + 3.46i·23-s − 8.30·25-s + (−0.340 + 0.592i)26-s + ⋯ |
| L(s) = 1 | + (−0.866 − 0.498i)2-s + (0.502 + 0.864i)4-s − 1.63i·5-s + (−0.00508 − 0.999i)8-s + (−0.813 + 1.41i)10-s + 1.83i·11-s − 0.134i·13-s + (−0.494 + 0.869i)16-s − 0.619i·17-s − 0.279·19-s + (1.40 − 0.820i)20-s + (0.912 − 1.58i)22-s + 0.722i·23-s − 1.66·25-s + (−0.0668 + 0.116i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2236125589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2236125589\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.22 + 0.705i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.64iT - 5T^{2} \) |
| 11 | \( 1 - 6.07iT - 11T^{2} \) |
| 13 | \( 1 + 0.483iT - 13T^{2} \) |
| 17 | \( 1 + 2.55iT - 17T^{2} \) |
| 19 | \( 1 + 1.21T + 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 8.21T + 29T^{2} \) |
| 31 | \( 1 + 6.31T + 31T^{2} \) |
| 37 | \( 1 - 1.19T + 37T^{2} \) |
| 41 | \( 1 - 6.59iT - 41T^{2} \) |
| 43 | \( 1 - 3.51iT - 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 2.62T + 53T^{2} \) |
| 59 | \( 1 + 1.16T + 59T^{2} \) |
| 61 | \( 1 + 0.208iT - 61T^{2} \) |
| 67 | \( 1 + 1.77iT - 67T^{2} \) |
| 71 | \( 1 + 9.13iT - 71T^{2} \) |
| 73 | \( 1 + 6.04iT - 73T^{2} \) |
| 79 | \( 1 + 3.17iT - 79T^{2} \) |
| 83 | \( 1 - 7.49T + 83T^{2} \) |
| 89 | \( 1 - 12.2iT - 89T^{2} \) |
| 97 | \( 1 - 11.8iT - 97T^{2} \) |
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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
−9.396366017500344345330522749685, −9.062029329204594850645833622585, −7.87243492303805294435258305563, −7.60518167918710109282489267359, −6.51116165335167284020621777969, −5.20471774205448206603918551634, −4.57503895678643070633890973242, −3.60112322637437161234151564784, −2.09407585733449697580226930526, −1.39262584169717754347452067075,
0.10926513226103249788606553191, 1.89047704649489461146404144516, 2.96856292449718917683380058082, 3.80461896529653251321057204412, 5.51304478636564709102734737993, 6.07994313733452296690389050395, 6.76346494248409967602691780167, 7.47158461421429963500794549100, 8.309243471745594888971296788131, 8.946044360528562660308283554763
