L(s) = 1 | + (−0.334 + 0.942i)2-s + (0.203 − 0.979i)3-s + (−0.775 − 0.631i)4-s + (−0.0682 − 0.997i)5-s + (0.854 + 0.519i)6-s + (−0.576 − 0.816i)7-s + (0.854 − 0.519i)8-s + (−0.917 − 0.398i)9-s + (0.962 + 0.269i)10-s + (0.460 + 0.887i)11-s + (−0.775 + 0.631i)12-s + (0.682 + 0.730i)13-s + (0.962 − 0.269i)14-s + (−0.990 − 0.136i)15-s + (0.203 + 0.979i)16-s + (0.460 − 0.887i)17-s + ⋯ |
L(s) = 1 | + (−0.334 + 0.942i)2-s + (0.203 − 0.979i)3-s + (−0.775 − 0.631i)4-s + (−0.0682 − 0.997i)5-s + (0.854 + 0.519i)6-s + (−0.576 − 0.816i)7-s + (0.854 − 0.519i)8-s + (−0.917 − 0.398i)9-s + (0.962 + 0.269i)10-s + (0.460 + 0.887i)11-s + (−0.775 + 0.631i)12-s + (0.682 + 0.730i)13-s + (0.962 − 0.269i)14-s + (−0.990 − 0.136i)15-s + (0.203 + 0.979i)16-s + (0.460 − 0.887i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6326664467 - 0.2327860540i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6326664467 - 0.2327860540i\) |
\(L(1)\) |
\(\approx\) |
\(0.8039000200 - 0.1099745280i\) |
\(L(1)\) |
\(\approx\) |
\(0.8039000200 - 0.1099745280i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 47 | \( 1 \) |
good | 2 | \( 1 + (-0.334 + 0.942i)T \) |
| 3 | \( 1 + (0.203 - 0.979i)T \) |
| 5 | \( 1 + (-0.0682 - 0.997i)T \) |
| 7 | \( 1 + (-0.576 - 0.816i)T \) |
| 11 | \( 1 + (0.460 + 0.887i)T \) |
| 13 | \( 1 + (0.682 + 0.730i)T \) |
| 17 | \( 1 + (0.460 - 0.887i)T \) |
| 19 | \( 1 + (-0.0682 + 0.997i)T \) |
| 23 | \( 1 + (-0.334 - 0.942i)T \) |
| 29 | \( 1 + (0.682 - 0.730i)T \) |
| 31 | \( 1 + (0.203 + 0.979i)T \) |
| 37 | \( 1 + (0.962 + 0.269i)T \) |
| 41 | \( 1 + (0.854 + 0.519i)T \) |
| 43 | \( 1 + (-0.775 - 0.631i)T \) |
| 53 | \( 1 + (0.854 + 0.519i)T \) |
| 59 | \( 1 + (-0.775 + 0.631i)T \) |
| 61 | \( 1 + (0.962 - 0.269i)T \) |
| 67 | \( 1 + (-0.576 + 0.816i)T \) |
| 71 | \( 1 + (-0.334 - 0.942i)T \) |
| 73 | \( 1 + (-0.917 + 0.398i)T \) |
| 79 | \( 1 + (-0.990 - 0.136i)T \) |
| 83 | \( 1 + (0.460 + 0.887i)T \) |
| 89 | \( 1 + (-0.0682 - 0.997i)T \) |
| 97 | \( 1 + (0.203 - 0.979i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−34.51147465094972359748732368144, −32.760781229692999486521836283736, −31.82188912910045191177205320276, −30.67744644315947629632643606160, −29.6057184276464476523169661561, −28.213437616076975584498343396585, −27.44365820290732572973960408404, −26.235908868856665585487369229425, −25.54936302788395545081177093726, −23.05489785182110166614386788059, −21.93335221538390019009129126981, −21.524878731533080764571581360157, −19.839916443687986850117100351903, −19.00284855262403129670040457427, −17.67172817403053854640932957469, −16.08952829077632371852024274669, −14.81139094347824581923353911122, −13.38966660323227343857106426805, −11.58744958270362759188400866611, −10.65406162845602854076056350583, −9.478507945659891378043997964799, −8.27614044477031593576035137565, −5.84249747968992863564941298874, −3.70600184705884017134004188185, −2.80478196950694983891548706542,
1.20958362185050350126488009537, 4.307541438980147409695869141810, 6.17399511417007680228406114264, 7.32270987816231367064136319505, 8.556454199789278781152701148754, 9.80989198606172332452812591692, 12.16302091601027197968670170443, 13.38427642439584059080447994407, 14.341286060551002932114692037118, 16.19425373473041721418838056921, 17.00282529797616187990911749883, 18.26108788796643636208692210024, 19.503011553064567739137823929643, 20.50810451223241721895781782846, 22.96662832566682865357390626328, 23.49152117146144436114052541581, 24.81287926721951438592547610386, 25.44860144795558328474840987375, 26.74481722940798595212137337263, 28.209679908163127429109436440071, 29.111929773805533412141216104794, 30.69019539368123080176358870600, 31.85885384513593082018709513865, 32.82792716665669938715484134687, 34.01216453273564407411225377556