L(s) = 1 | + (−0.786 + 0.618i)2-s + (−0.5 + 0.866i)3-s + (0.235 − 0.971i)4-s + (−0.142 − 0.989i)6-s + (0.415 + 0.909i)8-s + (−0.5 − 0.866i)9-s + (0.723 + 0.690i)12-s + (0.959 − 0.281i)13-s + (−0.888 − 0.458i)16-s + (−0.981 + 0.189i)17-s + (0.928 + 0.371i)18-s + (0.981 + 0.189i)19-s + (0.888 + 0.458i)23-s + (−0.995 − 0.0950i)24-s + (−0.580 + 0.814i)26-s + 27-s + ⋯ |
L(s) = 1 | + (−0.786 + 0.618i)2-s + (−0.5 + 0.866i)3-s + (0.235 − 0.971i)4-s + (−0.142 − 0.989i)6-s + (0.415 + 0.909i)8-s + (−0.5 − 0.866i)9-s + (0.723 + 0.690i)12-s + (0.959 − 0.281i)13-s + (−0.888 − 0.458i)16-s + (−0.981 + 0.189i)17-s + (0.928 + 0.371i)18-s + (0.981 + 0.189i)19-s + (0.888 + 0.458i)23-s + (−0.995 − 0.0950i)24-s + (−0.580 + 0.814i)26-s + 27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8985850532 + 0.2380866996i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8985850532 + 0.2380866996i\) |
\(L(1)\) |
\(\approx\) |
\(0.6226901714 + 0.2583447130i\) |
\(L(1)\) |
\(\approx\) |
\(0.6226901714 + 0.2583447130i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.786 + 0.618i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.959 - 0.281i)T \) |
| 17 | \( 1 + (-0.981 + 0.189i)T \) |
| 19 | \( 1 + (0.981 + 0.189i)T \) |
| 23 | \( 1 + (0.888 + 0.458i)T \) |
| 29 | \( 1 + (0.654 - 0.755i)T \) |
| 31 | \( 1 + (-0.723 + 0.690i)T \) |
| 37 | \( 1 + (-0.235 - 0.971i)T \) |
| 41 | \( 1 + (-0.142 - 0.989i)T \) |
| 43 | \( 1 + (0.415 + 0.909i)T \) |
| 47 | \( 1 + (0.928 - 0.371i)T \) |
| 53 | \( 1 + (0.888 - 0.458i)T \) |
| 59 | \( 1 + (0.786 + 0.618i)T \) |
| 61 | \( 1 + (0.928 - 0.371i)T \) |
| 67 | \( 1 + (-0.928 - 0.371i)T \) |
| 71 | \( 1 + (-0.654 + 0.755i)T \) |
| 73 | \( 1 + (-0.0475 - 0.998i)T \) |
| 79 | \( 1 + (0.995 - 0.0950i)T \) |
| 83 | \( 1 + (-0.841 - 0.540i)T \) |
| 89 | \( 1 + (0.327 - 0.945i)T \) |
| 97 | \( 1 + (0.415 + 0.909i)T \) |
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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
−18.336030859058801440771883045031, −17.8306819711697144107798911592, −17.19181667367490291180778500298, −16.42854856371727865147261834313, −15.994618305714194409891038501302, −15.0462393171598510245881987142, −13.82589466212817078597682370987, −13.42975310886340380950384845622, −12.76393585308101058800940363727, −12.02983176196865179643646100301, −11.379167453287257601536537706060, −10.95681204721157583164895244164, −10.241499842157942890315763039848, −9.20449681213841480974527364379, −8.68304164760542968801866072172, −8.01023300761423810014220535344, −7.03906413168137581753867853352, −6.81400364537297223198397761326, −5.8228997923112287980316545648, −4.890227229017768625397639689205, −3.97714350455307019134812193109, −2.969698893561175825090530278039, −2.33539275756207385338345804185, −1.36181535240891678200430667747, −0.82067613233917777941846808459,
0.51992313278867627668497396690, 1.40011396094869004426996936312, 2.57523821763748952158322256033, 3.59632748204669388128150790725, 4.37560524432146968936316078946, 5.35383778431734115681798820420, 5.69168654183971892434791470600, 6.57946941383461750594641158693, 7.20654361983751540356730404680, 8.163134893210614901058812718397, 9.00242917712072719034216383852, 9.21430687095022996706263482410, 10.28281070911928348719331122953, 10.66347344151883486228152622480, 11.347355251538868090965261098919, 11.9673148282503159904546564617, 13.135389471630927823968097467025, 13.86450804611615143310969569822, 14.66700156310121934428979284238, 15.29842840445923786322921316925, 15.98764302166020597885528403739, 16.16353646711018087458528123029, 17.1304540390234862852092168581, 17.75119058042597131325524360436, 18.04347121691699947904574150714