L(s) = 1 | + (0.984 + 0.173i)2-s + (0.893 − 0.448i)3-s + (0.939 + 0.342i)4-s + (0.396 − 0.918i)5-s + (0.957 − 0.286i)6-s + (−0.993 + 0.116i)7-s + (0.866 + 0.5i)8-s + (0.597 − 0.802i)9-s + (0.549 − 0.835i)10-s + (0.549 + 0.835i)11-s + (0.993 − 0.116i)12-s + (0.116 + 0.993i)13-s + (−0.998 − 0.0581i)14-s + (−0.0581 − 0.998i)15-s + (0.766 + 0.642i)16-s + (−0.642 + 0.766i)17-s + ⋯ |
L(s) = 1 | + (0.984 + 0.173i)2-s + (0.893 − 0.448i)3-s + (0.939 + 0.342i)4-s + (0.396 − 0.918i)5-s + (0.957 − 0.286i)6-s + (−0.993 + 0.116i)7-s + (0.866 + 0.5i)8-s + (0.597 − 0.802i)9-s + (0.549 − 0.835i)10-s + (0.549 + 0.835i)11-s + (0.993 − 0.116i)12-s + (0.116 + 0.993i)13-s + (−0.998 − 0.0581i)14-s + (−0.0581 − 0.998i)15-s + (0.766 + 0.642i)16-s + (−0.642 + 0.766i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.154818056 + 3.078815139i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.154818056 + 3.078815139i\) |
\(L(1)\) |
\(\approx\) |
\(2.308725529 + 0.1126951000i\) |
\(L(1)\) |
\(\approx\) |
\(2.308725529 + 0.1126951000i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 3 | \( 1 + (0.893 - 0.448i)T \) |
| 5 | \( 1 + (0.396 - 0.918i)T \) |
| 7 | \( 1 + (-0.993 + 0.116i)T \) |
| 11 | \( 1 + (0.549 + 0.835i)T \) |
| 13 | \( 1 + (0.116 + 0.993i)T \) |
| 17 | \( 1 + (-0.642 + 0.766i)T \) |
| 19 | \( 1 + (0.642 - 0.766i)T \) |
| 23 | \( 1 + (-0.984 + 0.173i)T \) |
| 29 | \( 1 + (-0.893 + 0.448i)T \) |
| 31 | \( 1 + (-0.597 + 0.802i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (-0.766 - 0.642i)T \) |
| 47 | \( 1 + (0.230 + 0.973i)T \) |
| 53 | \( 1 + (-0.802 + 0.597i)T \) |
| 59 | \( 1 + (-0.998 + 0.0581i)T \) |
| 61 | \( 1 + (0.686 - 0.727i)T \) |
| 67 | \( 1 + (0.918 - 0.396i)T \) |
| 71 | \( 1 + (-0.173 - 0.984i)T \) |
| 73 | \( 1 + (-0.835 + 0.549i)T \) |
| 79 | \( 1 + (-0.116 + 0.993i)T \) |
| 83 | \( 1 + (-0.893 + 0.448i)T \) |
| 89 | \( 1 + (-0.686 + 0.727i)T \) |
| 97 | \( 1 + (0.396 - 0.918i)T \) |
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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
−18.53303971758530667907142280167, −17.26254068335027729368531465899, −16.26725552779119490455474857725, −15.97400813202432643231966012453, −15.18088586313862164485073153757, −14.59786944489291718689524712484, −13.96118961977362504740465162457, −13.43104554006420277970274788304, −13.00640064479159499125842489091, −11.92863446191861544434144914015, −11.18020959151737784565797229789, −10.47473691946724876305581876690, −9.877012415498265002968887112028, −9.37540805163053405439520779406, −8.22569016750608836890984102589, −7.40442565819476194295412647270, −6.805543241603382515857852845248, −5.87373049953672634596204712228, −5.53159805599708140782366783055, −4.1876887608591357936772330262, −3.595535009921497164020677963143, −3.13008902744087690710393371760, −2.47066824264262344170332107511, −1.64134215665362313254967701309, −0.25179832398624235183178871015,
1.37955650791340184576734605970, 1.822521956719281918942898600871, 2.6377679678527721655550253371, 3.61428715894343152645692315744, 4.14210400623536823129831660769, 4.88498595589331247295941364983, 5.922095127712144520884459660019, 6.584053575196440683568172385258, 7.060916725113604231221117555316, 7.94832487557005984408099972638, 8.852942747608934433010691121572, 9.373984911161459255789243195655, 9.98401552431009752208599850333, 11.20626183751498436212522328069, 12.08196879610094712435529503242, 12.61997289477447371533855682866, 12.98784179837610423287816305206, 13.80356634973358014841895453174, 14.13304555114642634230504282032, 15.07851919151283853696281379478, 15.66957292951027618161229343010, 16.26985188747891665472970494162, 17.00156046555011166622703461870, 17.712846098651021522684141276131, 18.626505081399964656660189877532