L(s) = 1 | + (0.241 − 0.970i)2-s + (0.615 − 0.788i)3-s + (−0.882 − 0.469i)4-s + (−0.615 − 0.788i)6-s + (−0.5 + 0.866i)7-s + (−0.669 + 0.743i)8-s + (−0.241 − 0.970i)9-s + (0.913 + 0.406i)11-s + (−0.913 + 0.406i)12-s + (−0.961 − 0.275i)13-s + (0.719 + 0.694i)14-s + (0.559 + 0.829i)16-s + (0.848 + 0.529i)17-s − 18-s + (0.374 + 0.927i)21-s + (0.615 − 0.788i)22-s + ⋯ |
L(s) = 1 | + (0.241 − 0.970i)2-s + (0.615 − 0.788i)3-s + (−0.882 − 0.469i)4-s + (−0.615 − 0.788i)6-s + (−0.5 + 0.866i)7-s + (−0.669 + 0.743i)8-s + (−0.241 − 0.970i)9-s + (0.913 + 0.406i)11-s + (−0.913 + 0.406i)12-s + (−0.961 − 0.275i)13-s + (0.719 + 0.694i)14-s + (0.559 + 0.829i)16-s + (0.848 + 0.529i)17-s − 18-s + (0.374 + 0.927i)21-s + (0.615 − 0.788i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.968189485 - 1.035540621i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.968189485 - 1.035540621i\) |
\(L(1)\) |
\(\approx\) |
\(1.096814629 - 0.6953335436i\) |
\(L(1)\) |
\(\approx\) |
\(1.096814629 - 0.6953335436i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.241 - 0.970i)T \) |
| 3 | \( 1 + (0.615 - 0.788i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.913 + 0.406i)T \) |
| 13 | \( 1 + (-0.961 - 0.275i)T \) |
| 17 | \( 1 + (0.848 + 0.529i)T \) |
| 23 | \( 1 + (0.438 + 0.898i)T \) |
| 29 | \( 1 + (-0.848 + 0.529i)T \) |
| 31 | \( 1 + (0.978 + 0.207i)T \) |
| 37 | \( 1 + (0.809 + 0.587i)T \) |
| 41 | \( 1 + (-0.559 - 0.829i)T \) |
| 43 | \( 1 + (0.173 + 0.984i)T \) |
| 47 | \( 1 + (0.848 - 0.529i)T \) |
| 53 | \( 1 + (0.882 + 0.469i)T \) |
| 59 | \( 1 + (0.997 - 0.0697i)T \) |
| 61 | \( 1 + (0.438 + 0.898i)T \) |
| 67 | \( 1 + (0.374 - 0.927i)T \) |
| 71 | \( 1 + (-0.990 - 0.139i)T \) |
| 73 | \( 1 + (0.961 - 0.275i)T \) |
| 79 | \( 1 + (0.615 - 0.788i)T \) |
| 83 | \( 1 + (-0.978 - 0.207i)T \) |
| 89 | \( 1 + (-0.559 + 0.829i)T \) |
| 97 | \( 1 + (0.374 + 0.927i)T \) |
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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
−23.82117822871398923921835541722, −22.70495396068069363311628534970, −22.311373637466900795824547794552, −21.33343865380841874677665485270, −20.44611950605549626962917949502, −19.43792273664154510238500368684, −18.74740972802602334602167311252, −17.143516881430302035369815316735, −16.789210413008971597363236215179, −16.072698113913606331507818037689, −14.99065373454633408143814980544, −14.32213800185852521471562283524, −13.73626907606431871183213714324, −12.71193848620783173009783706749, −11.483240145029992988582803476047, −10.05680248577401651529484458587, −9.55141816469414917683066199018, −8.57894675405669812442840096819, −7.56662853646179098534447483439, −6.77794485556097202314799306555, −5.5654943799271568272111070110, −4.456655714083927781706195584494, −3.8120189978491800276752197420, −2.7553934041134106476976986422, −0.60244506285808636745005588617,
0.99329668779134148713908495540, 2.04029916443044728931004813113, 2.949230952740299721890512265263, 3.81947706261358377901134852487, 5.26035613849506677542693936113, 6.23398732251900491182163099098, 7.42746396689653911131444505427, 8.59415110532965869516990214044, 9.37602142976270947475710043535, 10.054959104838749903834683435, 11.611894963251327479708905648746, 12.19653317757780723251034004679, 12.81367507053126751436568140226, 13.73474171435347608512550193652, 14.77975822209492849966009854879, 15.13008298214807502187504456441, 16.9359268068046432405429803645, 17.77593588295319843203598593771, 18.70451110653983860615065933051, 19.37121564711594391411821962056, 19.8400736806172645836010816899, 20.84617806995610181953115113131, 21.76021832224481114602167220784, 22.498116009718116568633603667664, 23.3682685577051536987630453778