L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 + 0.984i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (0.766 + 0.642i)6-s + (0.173 + 0.984i)7-s + (−0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.766 + 0.642i)10-s + (0.766 − 0.642i)11-s + 12-s + (−0.939 − 0.342i)13-s + (0.766 + 0.642i)14-s + (−0.939 + 0.342i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)2-s + (0.173 + 0.984i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (0.766 + 0.642i)6-s + (0.173 + 0.984i)7-s + (−0.5 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.766 + 0.642i)10-s + (0.766 − 0.642i)11-s + 12-s + (−0.939 − 0.342i)13-s + (0.766 + 0.642i)14-s + (−0.939 + 0.342i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5990901559 + 1.565695653i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5990901559 + 1.565695653i\) |
\(L(1)\) |
\(\approx\) |
\(1.411112560 + 0.3382256650i\) |
\(L(1)\) |
\(\approx\) |
\(1.411112560 + 0.3382256650i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (0.173 + 0.984i)T \) |
| 5 | \( 1 + (0.173 + 0.984i)T \) |
| 7 | \( 1 + (0.173 + 0.984i)T \) |
| 11 | \( 1 + (0.766 - 0.642i)T \) |
| 13 | \( 1 + (-0.939 - 0.342i)T \) |
| 17 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (0.173 - 0.984i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.939 + 0.342i)T \) |
| 31 | \( 1 + (-0.939 - 0.342i)T \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.173 + 0.984i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.939 + 0.342i)T \) |
| 61 | \( 1 + (0.173 + 0.984i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.766 + 0.642i)T \) |
| 73 | \( 1 + (0.173 + 0.984i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 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
−17.97330069976598906036501904192, −17.203566136999657182424581689660, −16.890799999944189944859015106328, −16.47382618066163619512772350457, −15.32240514412143582953915640478, −14.54309299175956527070032305469, −14.06024294611772892165255225411, −13.53683826090795014662581643753, −12.77187365910043742076302259630, −12.25016417657477951151620994712, −11.80240418711176135123823180320, −10.88478761124247609187604353951, −9.50499892082996953884494336788, −9.14887207542506087009631924469, −8.148172081202125032777254249487, −7.3723767082122831068812735072, −7.21024770850818895501562502702, −6.30949020660552904659640162573, −5.38915219037081204723342336897, −4.84828116797364488798927693050, −4.01160676280153065428299427754, −3.28311714016149634664750505989, −2.08305324806766418629304208611, −1.522863777145352621563578091417, −0.30791412959666226364013509060,
1.43618819853228159353640297152, 2.511090616382317272592538304265, 2.88761206813933813465778395792, 3.59655739569039628364724770467, 4.39397217145732665313676605525, 5.282807272350255298157639736930, 5.77142109602347797698377981343, 6.48178126186447489169869138068, 7.447481178935185384891109439040, 8.59084369861392582774046202361, 9.36591636725919456965772027269, 9.72994814629372588327029803481, 10.68880393784958919525517671519, 11.26147512824441412677374083143, 11.51945933826973245033694588244, 12.62002592313334703267131859505, 13.224884484814545974457739835912, 14.34448091596958824630140374711, 14.58833353413648089259668474751, 15.08190783204620800536063235159, 15.62913049113243674766997647744, 16.582952770883277018932450102646, 17.38640774929700308217112588699, 18.17440781517453745667668859030, 19.07069337107192555553686077501