L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.686 + 0.727i)3-s + (−0.939 − 0.342i)4-s + (0.893 − 0.448i)5-s + (0.597 + 0.802i)6-s + (−0.835 − 0.549i)7-s + (−0.5 + 0.866i)8-s + (−0.0581 − 0.998i)9-s + (−0.286 − 0.957i)10-s + (−0.286 + 0.957i)11-s + (0.893 − 0.448i)12-s + (−0.835 − 0.549i)13-s + (−0.686 + 0.727i)14-s + (−0.286 + 0.957i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.686 + 0.727i)3-s + (−0.939 − 0.342i)4-s + (0.893 − 0.448i)5-s + (0.597 + 0.802i)6-s + (−0.835 − 0.549i)7-s + (−0.5 + 0.866i)8-s + (−0.0581 − 0.998i)9-s + (−0.286 − 0.957i)10-s + (−0.286 + 0.957i)11-s + (0.893 − 0.448i)12-s + (−0.835 − 0.549i)13-s + (−0.686 + 0.727i)14-s + (−0.286 + 0.957i)15-s + (0.766 + 0.642i)16-s + (0.766 + 0.642i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.077605743 - 0.4610673386i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.077605743 - 0.4610673386i\) |
\(L(1)\) |
\(\approx\) |
\(0.8007154304 - 0.3280294569i\) |
\(L(1)\) |
\(\approx\) |
\(0.8007154304 - 0.3280294569i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (-0.686 + 0.727i)T \) |
| 5 | \( 1 + (0.893 - 0.448i)T \) |
| 7 | \( 1 + (-0.835 - 0.549i)T \) |
| 11 | \( 1 + (-0.286 + 0.957i)T \) |
| 13 | \( 1 + (-0.835 - 0.549i)T \) |
| 17 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + (0.173 - 0.984i)T \) |
| 23 | \( 1 + (0.766 - 0.642i)T \) |
| 29 | \( 1 + (0.973 + 0.230i)T \) |
| 31 | \( 1 + (0.893 + 0.448i)T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.939 + 0.342i)T \) |
| 47 | \( 1 + (-0.993 - 0.116i)T \) |
| 53 | \( 1 + (-0.835 - 0.549i)T \) |
| 59 | \( 1 + (-0.286 + 0.957i)T \) |
| 61 | \( 1 + (0.597 + 0.802i)T \) |
| 67 | \( 1 + (-0.0581 - 0.998i)T \) |
| 71 | \( 1 + (0.173 + 0.984i)T \) |
| 73 | \( 1 + (-0.286 + 0.957i)T \) |
| 79 | \( 1 + (0.893 + 0.448i)T \) |
| 83 | \( 1 + (-0.286 - 0.957i)T \) |
| 89 | \( 1 + (-0.993 + 0.116i)T \) |
| 97 | \( 1 + (-0.835 - 0.549i)T \) |
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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
−18.50618248675131713971083403240, −17.79429262214000770470369269159, −17.06862364843258686587557519535, −16.62103204511329384527986667202, −16.04286339459832134302928032126, −15.20879318029952260212801039018, −14.29231269264581007411160470146, −13.76022252828477949809386242008, −13.34606739764094935448683774641, −12.419555398945822682340687806423, −12.02794437368172301971835552189, −11.04984859969359293437516124640, −9.857244910102676319069134875030, −9.77769540148653739130865580829, −8.65142833522425376866747680337, −7.91775952554893816632838104809, −7.07300324474925391977844877723, −6.55459073118152666073015965194, −5.94350609191353023564198890505, −5.40988007853481966963508402158, −4.80902663842418220088331867177, −3.338101646411187148511351597549, −2.83580971824171279977291254446, −1.7134681698746395571618299347, −0.54357298030200585485278666197,
0.67496200671275865569975689059, 1.42360691870948233481338862443, 2.71813536316550607909520767427, 3.10239800467779551446905756785, 4.260293848448906543582610876715, 4.90079307040657150503778517177, 5.22828339855836773817117586940, 6.29198001018342536458814090358, 6.86290313268902365396554195407, 8.1884742147668260531550972559, 9.07354771176590277982781668822, 9.80186500786002362924294565117, 10.13066261148560071169825422721, 10.48164019281801356989660130943, 11.520227335077729167698405208135, 12.33971728422065245731614287805, 12.77622258777765117520748285040, 13.27288673211538130948805096382, 14.264387478099586964228211220277, 14.89195495117898844565276777192, 15.63164268090270966668221403506, 16.64404205548635748331246245685, 17.08837173880782358825694189604, 17.71407913582606475887997559771, 18.16233136795141134979657762852