L(s) = 1 | − 2-s + (0.286 − 0.957i)3-s + 4-s + (0.448 − 0.893i)5-s + (−0.286 + 0.957i)6-s + (−0.835 + 0.549i)7-s − 8-s + (−0.835 − 0.549i)9-s + (−0.448 + 0.893i)10-s + (0.448 + 0.893i)11-s + (0.286 − 0.957i)12-s + (−0.893 − 0.448i)13-s + (0.835 − 0.549i)14-s + (−0.727 − 0.686i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | − 2-s + (0.286 − 0.957i)3-s + 4-s + (0.448 − 0.893i)5-s + (−0.286 + 0.957i)6-s + (−0.835 + 0.549i)7-s − 8-s + (−0.835 − 0.549i)9-s + (−0.448 + 0.893i)10-s + (0.448 + 0.893i)11-s + (0.286 − 0.957i)12-s + (−0.893 − 0.448i)13-s + (0.835 − 0.549i)14-s + (−0.727 − 0.686i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0463 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0463 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5665460261 - 0.5934351091i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5665460261 - 0.5934351091i\) |
\(L(1)\) |
\(\approx\) |
\(0.6196863850 - 0.2544659692i\) |
\(L(1)\) |
\(\approx\) |
\(0.6196863850 - 0.2544659692i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.286 - 0.957i)T \) |
| 5 | \( 1 + (0.448 - 0.893i)T \) |
| 7 | \( 1 + (-0.835 + 0.549i)T \) |
| 11 | \( 1 + (0.448 + 0.893i)T \) |
| 13 | \( 1 + (-0.893 - 0.448i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.766 - 0.642i)T \) |
| 23 | \( 1 + (-0.939 + 0.342i)T \) |
| 29 | \( 1 + (0.549 + 0.835i)T \) |
| 31 | \( 1 + (0.802 - 0.597i)T \) |
| 41 | \( 1 + (-0.342 + 0.939i)T \) |
| 43 | \( 1 + (0.642 + 0.766i)T \) |
| 47 | \( 1 + (0.957 - 0.286i)T \) |
| 53 | \( 1 + (0.230 + 0.973i)T \) |
| 59 | \( 1 + (0.686 - 0.727i)T \) |
| 61 | \( 1 + (0.918 - 0.396i)T \) |
| 67 | \( 1 + (0.957 - 0.286i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (0.835 + 0.549i)T \) |
| 79 | \( 1 + (0.686 - 0.727i)T \) |
| 83 | \( 1 + (0.993 + 0.116i)T \) |
| 89 | \( 1 + (0.998 - 0.0581i)T \) |
| 97 | \( 1 + (-0.802 - 0.597i)T \) |
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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
−18.91822659089807699550883109834, −17.82614084217897358920152623610, −17.185157515706750937160486744973, −16.69045921436951563140256552475, −15.98063060930822407920719189824, −15.52579178820964853608529941794, −14.54617300915157601350735957927, −14.120260851345967675092288640894, −13.45702176589555812814720886018, −12.09944088803994744424509046973, −11.59279180705767963176443760310, −10.57105807458865443737165816228, −10.39000827429930501544114733794, −9.69162025113969413935935158892, −9.1347889145567792855621331037, −8.39375706328852671489361953767, −7.51677689677652528441429665797, −6.68563345893209252003831366087, −6.25216468000985731648907486068, −5.375513824176802400691719598002, −4.07221453532270903501579678162, −3.54498338553143221473197988853, −2.51566768807339306966734866662, −2.28926200077625696617570026119, −0.643002446078703396826303341501,
0.45919629287857056523763106014, 1.439522878465478321973181745315, 2.25222907265748501502146919245, 2.58909045852899516028996651273, 3.82732905482363230890482653876, 4.98013084898296055287940799998, 6.00372067568885348343696178348, 6.4136687034639950508260127749, 7.12563518299471754785435434953, 8.01559528236196576591103075498, 8.538628881087051197351950372577, 9.26890892738762561820392602508, 9.70728197412383083506582050877, 10.46310675925441947584057598931, 11.62949726030647229004168404914, 12.25554492506381897894187686773, 12.668185897393415091240034591437, 13.1930095881994130450134017881, 14.270119034672590740236345800199, 15.098334198784903551890233724095, 15.568251606659312522052180461289, 16.54125035415617291315675184287, 17.201171686058840037171043544196, 17.63179094165919300473001986494, 18.1087808893120536417205011344