L(s) = 1 | + (0.766 − 0.642i)2-s + (−0.766 + 0.642i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (−0.173 + 0.984i)6-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.766 − 0.642i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.173 + 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + (−0.5 − 0.866i)18-s + ⋯ |
L(s) = 1 | + (0.766 − 0.642i)2-s + (−0.766 + 0.642i)3-s + (0.173 − 0.984i)4-s + (−0.173 − 0.984i)5-s + (−0.173 + 0.984i)6-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.766 − 0.642i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (−0.173 + 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + (−0.5 − 0.866i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1160662268 - 0.3839051652i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1160662268 - 0.3839051652i\) |
\(L(1)\) |
\(\approx\) |
\(0.7991376746 - 0.3857221154i\) |
\(L(1)\) |
\(\approx\) |
\(0.7991376746 - 0.3857221154i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 5 | \( 1 + (-0.173 - 0.984i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.173 + 0.984i)T \) |
| 17 | \( 1 + (-0.173 - 0.984i)T \) |
| 23 | \( 1 + (-0.939 + 0.342i)T \) |
| 29 | \( 1 + (-0.939 + 0.342i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.766 - 0.642i)T \) |
| 47 | \( 1 + (-0.173 + 0.984i)T \) |
| 53 | \( 1 + (0.173 - 0.984i)T \) |
| 59 | \( 1 + (-0.173 - 0.984i)T \) |
| 61 | \( 1 + (0.939 - 0.342i)T \) |
| 67 | \( 1 + (0.766 + 0.642i)T \) |
| 71 | \( 1 + (0.766 - 0.642i)T \) |
| 73 | \( 1 + (-0.766 + 0.642i)T \) |
| 79 | \( 1 + (-0.939 - 0.342i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (-0.766 - 0.642i)T \) |
| 97 | \( 1 + (0.939 + 0.342i)T \) |
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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
−29.44650948552911720750836555085, −27.9701051843581892639212334823, −26.7204638794773936233004500602, −25.870472912885255400231664485422, −24.68606757588535303792996169670, −23.88869484123814533612760023301, −23.01829086845868131832444350555, −22.22338691831801828181053861636, −21.49402548866378318928403404470, −19.80919124419820250610633754448, −18.51227288301828826964491051873, −17.755024627216160433518554387134, −16.644299435739187130254604620613, −15.60196843691561156641983063365, −14.53846031407883647173186983991, −13.415846826405176984839462364966, −12.55205934707873678260086177990, −11.33641250060094452486051868163, −10.52482945341818127999897460721, −8.18646387183121969569979029424, −7.39658512771873769448781248262, −6.18922394014544260536510712691, −5.52142377209636138602054478072, −3.79903104689256804590742088464, −2.41713382455092040900885285255,
0.12692743970062614654138039927, 1.84379502018486748105635780480, 3.82968022963815236118652541769, 4.76449726738890858859840249336, 5.54003048415216480868350935719, 7.0586586960674846562468218379, 9.17694036735985577062784878812, 9.93294722644051109267108181411, 11.28323217430796255464311163725, 12.06731593242770771439233677099, 12.93329675994149038508327997586, 14.27909423531060880421982915933, 15.58215231944029875630731904490, 16.23324145926843481017217707711, 17.52052961317293413878609536186, 18.76668882493619095031349129475, 20.27937922734242081044221240380, 20.69628202743820139360722662197, 21.78397619748299701966130657148, 22.65729421645679157525237618636, 23.71249377157307325886613381235, 24.21837652541214020403949113628, 25.78094532281319408282184242302, 27.32421930391283771342410998316, 27.98022505270289503682475880258