L(s) = 1 | + (1.41 + i)3-s − 1.41i·5-s − i·7-s + (1.00 + 2.82i)9-s − 1.41·11-s + 6·13-s + (1.41 − 2.00i)15-s − 1.41i·17-s + 4i·19-s + (1 − 1.41i)21-s + 7.07·23-s + 2.99·25-s + (−1.41 + 5.00i)27-s − 8.48i·29-s + 2i·31-s + ⋯ |
L(s) = 1 | + (0.816 + 0.577i)3-s − 0.632i·5-s − 0.377i·7-s + (0.333 + 0.942i)9-s − 0.426·11-s + 1.66·13-s + (0.365 − 0.516i)15-s − 0.342i·17-s + 0.917i·19-s + (0.218 − 0.308i)21-s + 1.47·23-s + 0.599·25-s + (−0.272 + 0.962i)27-s − 1.57i·29-s + 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.07921 + 0.177074i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.07921 + 0.177074i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.41 - i)T \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 + 8.48iT - 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 16iT - 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 8.48T + 83T^{2} \) |
| 89 | \( 1 + 7.07iT - 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−10.54327170074998170260573861085, −9.542369054361820370214793422554, −8.752494283142902840414408899695, −8.190790495190963785841863847749, −7.21214845940047040670319965215, −5.91068444137787962800747837318, −4.85658650311411509883963892841, −3.94810555666117317244170380124, −2.97707788277338935080946401619, −1.37006641816021559330365390279,
1.40371970220097143396361952811, 2.84537587019504311731275778316, 3.48844988906674948765317015983, 5.01028731405641212035465919608, 6.34670161889675517148478419690, 6.88586182618558396721729149147, 7.938926182427047145610526652915, 8.773532716131252229045009760939, 9.304315613895865604555217203056, 10.73377374964354064374310128402