| L(s) = 1 | − 2-s + (1.72 + 0.163i)3-s + 4-s − 1.55i·5-s + (−1.72 − 0.163i)6-s + i·7-s − 8-s + (2.94 + 0.565i)9-s + 1.55i·10-s + (0.560 + 3.26i)11-s + (1.72 + 0.163i)12-s + 2.32i·13-s − i·14-s + (0.254 − 2.68i)15-s + 16-s + 6.13·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (0.995 + 0.0946i)3-s + 0.5·4-s − 0.695i·5-s + (−0.703 − 0.0669i)6-s + 0.377i·7-s − 0.353·8-s + (0.982 + 0.188i)9-s + 0.491i·10-s + (0.168 + 0.985i)11-s + (0.497 + 0.0473i)12-s + 0.645i·13-s − 0.267i·14-s + (0.0658 − 0.692i)15-s + 0.250·16-s + 1.48·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50745 + 0.0565001i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50745 + 0.0565001i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.72 - 0.163i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-0.560 - 3.26i)T \) |
| good | 5 | \( 1 + 1.55iT - 5T^{2} \) |
| 13 | \( 1 - 2.32iT - 13T^{2} \) |
| 17 | \( 1 - 6.13T + 17T^{2} \) |
| 19 | \( 1 + 3.69iT - 19T^{2} \) |
| 23 | \( 1 + 4.25iT - 23T^{2} \) |
| 29 | \( 1 + 1.46T + 29T^{2} \) |
| 31 | \( 1 + 1.87T + 31T^{2} \) |
| 37 | \( 1 + 6.88T + 37T^{2} \) |
| 41 | \( 1 - 2.23T + 41T^{2} \) |
| 43 | \( 1 + 12.1iT - 43T^{2} \) |
| 47 | \( 1 - 5.48iT - 47T^{2} \) |
| 53 | \( 1 - 9.89iT - 53T^{2} \) |
| 59 | \( 1 + 6.56iT - 59T^{2} \) |
| 61 | \( 1 - 7.35iT - 61T^{2} \) |
| 67 | \( 1 + 8.79T + 67T^{2} \) |
| 71 | \( 1 + 2.88iT - 71T^{2} \) |
| 73 | \( 1 - 1.67iT - 73T^{2} \) |
| 79 | \( 1 - 12.0iT - 79T^{2} \) |
| 83 | \( 1 + 0.338T + 83T^{2} \) |
| 89 | \( 1 + 5.55iT - 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−10.77409682061581171990395905313, −9.864250022772095091893472314656, −9.135547880469770501586012170770, −8.609757794393502498251754719288, −7.57150944899834742400840624695, −6.82619688968695308255741264174, −5.25999112398092846335988397660, −4.14531091395647296748801014606, −2.71287056837583306273975751563, −1.48952120035314651899274504074,
1.36759780211611231809443740567, 3.03378573041025157445093618364, 3.61142935466912550183365485510, 5.56074642222152577549358195670, 6.71917301459735585929390857725, 7.69088965510436039180669534277, 8.172043514898111414322383980019, 9.260861432257081545710251639461, 10.10465890307815239637782163568, 10.70185875680401005105588829081
