L(s) = 1 | + (0.984 − 0.173i)3-s + (0.642 + 0.766i)5-s + (−0.5 + 0.866i)7-s + (0.939 − 0.342i)9-s + (−0.866 + 0.5i)11-s + (0.984 + 0.173i)13-s + (0.766 + 0.642i)15-s + (−0.939 − 0.342i)17-s + (−0.342 + 0.939i)21-s + (0.766 + 0.642i)23-s + (−0.173 + 0.984i)25-s + (0.866 − 0.5i)27-s + (−0.342 − 0.939i)29-s + (−0.5 + 0.866i)31-s + (−0.766 + 0.642i)33-s + ⋯ |
L(s) = 1 | + (0.984 − 0.173i)3-s + (0.642 + 0.766i)5-s + (−0.5 + 0.866i)7-s + (0.939 − 0.342i)9-s + (−0.866 + 0.5i)11-s + (0.984 + 0.173i)13-s + (0.766 + 0.642i)15-s + (−0.939 − 0.342i)17-s + (−0.342 + 0.939i)21-s + (0.766 + 0.642i)23-s + (−0.173 + 0.984i)25-s + (0.866 − 0.5i)27-s + (−0.342 − 0.939i)29-s + (−0.5 + 0.866i)31-s + (−0.766 + 0.642i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.665942563 + 0.7101140658i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.665942563 + 0.7101140658i\) |
\(L(1)\) |
\(\approx\) |
\(1.455251384 + 0.3030374983i\) |
\(L(1)\) |
\(\approx\) |
\(1.455251384 + 0.3030374983i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.984 - 0.173i)T \) |
| 5 | \( 1 + (0.642 + 0.766i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| 13 | \( 1 + (0.984 + 0.173i)T \) |
| 17 | \( 1 + (-0.939 - 0.342i)T \) |
| 23 | \( 1 + (0.766 + 0.642i)T \) |
| 29 | \( 1 + (-0.342 - 0.939i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (0.173 + 0.984i)T \) |
| 43 | \( 1 + (-0.642 - 0.766i)T \) |
| 47 | \( 1 + (0.939 - 0.342i)T \) |
| 53 | \( 1 + (0.642 - 0.766i)T \) |
| 59 | \( 1 + (0.342 - 0.939i)T \) |
| 61 | \( 1 + (0.642 - 0.766i)T \) |
| 67 | \( 1 + (0.342 + 0.939i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (-0.173 - 0.984i)T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (0.173 - 0.984i)T \) |
| 97 | \( 1 + (0.939 + 0.342i)T \) |
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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
−25.42451836192978691211025515459, −24.29985540812806957297910532326, −23.72317832270208473221137814389, −22.3896980478559151288834941199, −21.35675750558869695735651156366, −20.56433206894956538185687712772, −20.12836919842209123954232793811, −18.97718105222184432388045727411, −18.07848089750745699496024933101, −16.81012191378266145611731059798, −16.1277263326332884316043518184, −15.18946015116584872113420768002, −13.899304845008223075680655334543, −13.267574453371218499326318268294, −12.8216737099246121482129272212, −10.89047641581364129107212780517, −10.20048129710271429024250413116, −9.04406732366989713307787596962, −8.449063761093098075319597523722, −7.26340187294027255357875887518, −6.039727220948316446409194313042, −4.72024021823477167102221897730, −3.69571549010758212143974610858, −2.528645007151535029662210969028, −1.15348044054989975030891533747,
1.91830575252523735422287660494, 2.66519109652193950042132103108, 3.682517555922208788895853669318, 5.31323818053432223759089837924, 6.48587699035500067507013302455, 7.334358933052589332999377128068, 8.63004472857910060530152765547, 9.38073215113362297181635303475, 10.301723286284484481465327921388, 11.45116981863038874769280538919, 12.95841611709111726181186334977, 13.34313410266638515136476701684, 14.45201345150319819115449015519, 15.37500962461973798135499419779, 15.90422977910187189170130908487, 17.63975368655502953592569039756, 18.40411406746446964242026488219, 18.93576590685486426691883759992, 20.03306981094624552504878369300, 21.07123511585388315758396854685, 21.63035245330188210554264775728, 22.717873087562999732292445085318, 23.65436004348195691562572639981, 25.08016508329509759145518767814, 25.27117012605425923699670150414