L(s) = 1 | + (0.0348 + 0.999i)2-s + (−0.997 + 0.0697i)4-s + (−0.939 − 0.342i)5-s + (−0.719 + 0.694i)7-s + (−0.104 − 0.994i)8-s + (0.309 − 0.951i)10-s + (−0.241 − 0.970i)11-s + (0.848 + 0.529i)13-s + (−0.719 − 0.694i)14-s + (0.990 − 0.139i)16-s + (−0.104 − 0.994i)17-s + (0.309 − 0.951i)19-s + (0.961 + 0.275i)20-s + (0.961 − 0.275i)22-s + (−0.241 + 0.970i)23-s + ⋯ |
L(s) = 1 | + (0.0348 + 0.999i)2-s + (−0.997 + 0.0697i)4-s + (−0.939 − 0.342i)5-s + (−0.719 + 0.694i)7-s + (−0.104 − 0.994i)8-s + (0.309 − 0.951i)10-s + (−0.241 − 0.970i)11-s + (0.848 + 0.529i)13-s + (−0.719 − 0.694i)14-s + (0.990 − 0.139i)16-s + (−0.104 − 0.994i)17-s + (0.309 − 0.951i)19-s + (0.961 + 0.275i)20-s + (0.961 − 0.275i)22-s + (−0.241 + 0.970i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2421464816 + 0.6354712002i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2421464816 + 0.6354712002i\) |
\(L(1)\) |
\(\approx\) |
\(0.6205484910 + 0.3651016659i\) |
\(L(1)\) |
\(\approx\) |
\(0.6205484910 + 0.3651016659i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (0.0348 + 0.999i)T \) |
| 5 | \( 1 + (-0.939 - 0.342i)T \) |
| 7 | \( 1 + (-0.719 + 0.694i)T \) |
| 11 | \( 1 + (-0.241 - 0.970i)T \) |
| 13 | \( 1 + (0.848 + 0.529i)T \) |
| 17 | \( 1 + (-0.104 - 0.994i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (-0.241 + 0.970i)T \) |
| 29 | \( 1 + (0.0348 + 0.999i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.374 - 0.927i)T \) |
| 43 | \( 1 + (0.0348 + 0.999i)T \) |
| 47 | \( 1 + (-0.374 + 0.927i)T \) |
| 53 | \( 1 + (-0.104 - 0.994i)T \) |
| 59 | \( 1 + (0.848 + 0.529i)T \) |
| 61 | \( 1 + (0.766 + 0.642i)T \) |
| 67 | \( 1 + (0.766 - 0.642i)T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (-0.104 + 0.994i)T \) |
| 79 | \( 1 + (-0.997 - 0.0697i)T \) |
| 83 | \( 1 + (-0.882 - 0.469i)T \) |
| 89 | \( 1 + (0.913 - 0.406i)T \) |
| 97 | \( 1 + (-0.719 + 0.694i)T \) |
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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
−21.92834372942449371592420071565, −20.62362792614997634999660769232, −20.377915768633607036915392282834, −19.50196776145353305089090312523, −18.85405105720290196736582154680, −18.10226217637881588477570345819, −17.18423157521837816158431859222, −16.21415503175567746760539036364, −15.28512374424345708764641116502, −14.49615975611034685237863835842, −13.49901273243222809007518071471, −12.67933681781051377569419356021, −12.16154151243082198392234849615, −11.07869189585560229205700591614, −10.37286416795554490574394427226, −9.87368702628863455475505759464, −8.52309040651647611838821917803, −7.89715566657202964942707669329, −6.81642959647845664773307379827, −5.70978144794749881190126694450, −4.314684541296810407429912271643, −3.85126143400103014579728270064, −2.997498957819756233001917837625, −1.76680773831816946819633066123, −0.402554737487827795615578135545,
0.96510048244790882481172706556, 3.02702219670584099433162004115, 3.6941627959776166899414759402, 4.87674077861423758031397051359, 5.61177873500504019648573163248, 6.630826372827835452080877085710, 7.342928248361212793983845177352, 8.495085007324121454291184207671, 8.86188274106509757705608342798, 9.76157005295833415474307926149, 11.20869507842774582144835509401, 11.85827678733567691828396974213, 12.988229098593929279168298714255, 13.52011869770915899414798640684, 14.49310769444496975077852693562, 15.63104410712211703872362365557, 15.947528705679731030732811696, 16.39572529026215329370225220234, 17.596736337763516622836482239253, 18.53756269530497470318206519719, 19.02237758194377038486943360976, 19.8302705829830453874605452741, 21.014937480072355155659492405342, 21.91170511808005618962913451432, 22.57510704942456927282371210157