L(s) = 1 | + (−0.642 − 0.766i)3-s + (0.342 − 0.939i)5-s + (−0.5 + 0.866i)7-s + (−0.173 + 0.984i)9-s + (−0.866 + 0.5i)11-s + (−0.642 + 0.766i)13-s + (−0.939 + 0.342i)15-s + (0.173 + 0.984i)17-s + (0.984 − 0.173i)21-s + (−0.939 + 0.342i)23-s + (−0.766 − 0.642i)25-s + (0.866 − 0.5i)27-s + (0.984 + 0.173i)29-s + (−0.5 + 0.866i)31-s + (0.939 + 0.342i)33-s + ⋯ |
L(s) = 1 | + (−0.642 − 0.766i)3-s + (0.342 − 0.939i)5-s + (−0.5 + 0.866i)7-s + (−0.173 + 0.984i)9-s + (−0.866 + 0.5i)11-s + (−0.642 + 0.766i)13-s + (−0.939 + 0.342i)15-s + (0.173 + 0.984i)17-s + (0.984 − 0.173i)21-s + (−0.939 + 0.342i)23-s + (−0.766 − 0.642i)25-s + (0.866 − 0.5i)27-s + (0.984 + 0.173i)29-s + (−0.5 + 0.866i)31-s + (0.939 + 0.342i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4041592548 + 0.3245890116i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4041592548 + 0.3245890116i\) |
\(L(1)\) |
\(\approx\) |
\(0.6842360550 + 0.02333321266i\) |
\(L(1)\) |
\(\approx\) |
\(0.6842360550 + 0.02333321266i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.642 - 0.766i)T \) |
| 5 | \( 1 + (0.342 - 0.939i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| 13 | \( 1 + (-0.642 + 0.766i)T \) |
| 17 | \( 1 + (0.173 + 0.984i)T \) |
| 23 | \( 1 + (-0.939 + 0.342i)T \) |
| 29 | \( 1 + (0.984 + 0.173i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (0.766 - 0.642i)T \) |
| 43 | \( 1 + (-0.342 + 0.939i)T \) |
| 47 | \( 1 + (-0.173 + 0.984i)T \) |
| 53 | \( 1 + (0.342 + 0.939i)T \) |
| 59 | \( 1 + (-0.984 + 0.173i)T \) |
| 61 | \( 1 + (0.342 + 0.939i)T \) |
| 67 | \( 1 + (-0.984 - 0.173i)T \) |
| 71 | \( 1 + (0.939 + 0.342i)T \) |
| 73 | \( 1 + (-0.766 + 0.642i)T \) |
| 79 | \( 1 + (0.766 - 0.642i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (0.766 + 0.642i)T \) |
| 97 | \( 1 + (-0.173 - 0.984i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−25.3644043118604300811773713924, −23.98657263246723491655997963793, −23.07949823601133316286964450930, −22.47677234508725507937531433354, −21.74030331256143455819245398463, −20.73633203454399260539366073146, −19.88311543405781264208337525819, −18.57406650565926748434159218298, −17.85625134956261672613377367550, −16.895393890055819055328541575080, −16.04689776832607981352333751483, −15.19148910976316703279458604011, −14.14825285344104990613558655764, −13.25317177068692493464048153860, −11.95144421580907732096617844858, −10.89629661116328368301914214831, −10.19594289417313530379034393704, −9.6657949922497979596234835978, −7.94359364614569769340642978538, −6.863951949048274134003224136493, −5.913136012755352935084529594061, −4.8849807518476842202297568286, −3.566852482077630003559092414856, −2.68743041741757264136598738848, −0.353781074422993059230314526914,
1.57578382860150414355312819461, 2.50643233648990383649402278906, 4.481415685953920766484920632778, 5.482927180768760426544374150931, 6.205107453866261012265585014, 7.47186553160965094750543411601, 8.4893759545224073428808293964, 9.54726189809818474539590540736, 10.60694123428999067242218260377, 12.08386546188761867618496880678, 12.42190457551860747549596501824, 13.23235041423768637603115600576, 14.36331640408108105994226446785, 15.80585198889832568148263298754, 16.41763757220873606021167253291, 17.50461345791367083253430453901, 18.107177094349404909541653375191, 19.23489313441821739797065076698, 19.86296255463843292625148575798, 21.34041676087218893436681065656, 21.783713484326970814685916127242, 23.02124636234643659576150402359, 23.8332143060605142903203281078, 24.5043060339051902045393214816, 25.37178942186033757296151633269