L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.309 + 0.951i)6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + 10-s + 12-s + (−0.809 − 0.587i)14-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.809 − 0.587i)18-s + (0.809 + 0.587i)19-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.309 + 0.951i)6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + 10-s + 12-s + (−0.809 − 0.587i)14-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.809 − 0.587i)18-s + (0.809 + 0.587i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5698154167 - 0.4037344713i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5698154167 - 0.4037344713i\) |
\(L(1)\) |
\(\approx\) |
\(0.6427676411 - 0.3253813563i\) |
\(L(1)\) |
\(\approx\) |
\(0.6427676411 - 0.3253813563i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.309 + 0.951i)T \) |
| 7 | \( 1 + (0.809 - 0.587i)T \) |
| 17 | \( 1 + (0.309 - 0.951i)T \) |
| 19 | \( 1 + (0.809 + 0.587i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
| 37 | \( 1 + (0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.809 + 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.809 + 0.587i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (0.809 - 0.587i)T \) |
| 61 | \( 1 + (0.309 - 0.951i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (-0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.809 - 0.587i)T \) |
| 79 | \( 1 + (0.309 + 0.951i)T \) |
| 83 | \( 1 + (-0.309 + 0.951i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.309 - 0.951i)T \) |
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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
−28.27603186044223160138425983686, −27.47758390018890014151837203793, −26.787544384493721726313862267182, −25.46289456349977735627696582706, −24.30207266666207563795740630974, −23.8622074226719266600134906071, −22.76549875407580270583275868640, −21.6725051075993441538072128126, −20.71483134828830015399600408386, −19.28221249331680857542034526313, −18.008092857705671110604650287494, −17.25317596256962873993493150659, −16.40819959547236579314522436553, −15.481683784276865483214503704187, −14.72816461992767832098733852549, −13.11914511386623082822632389575, −11.962372249114988763251694592790, −10.80293525538525275785206279303, −9.41027946698982536815297072972, −8.62508525538567492773861914203, −7.37766266848087566686055569667, −5.789462167175465275986398570665, −5.11618220995977758773736640410, −4.1000243457201644899913113598, −1.13377079501859907296241865843,
1.07073143529686900259349677684, 2.57094623460789239203138680876, 4.09491453265842737362613230934, 5.44404864388054589275363812804, 7.2487713873921671799985541104, 7.79862209008182826229652986500, 9.63524832989628723554368700351, 10.95912106231300727027583272664, 11.23986133233471211668600827512, 12.36631555189273506634884720344, 13.61418501803988404028334707107, 14.51148304798762792829667378490, 16.34659193602986494044520632973, 17.39355860924971232860125996685, 18.288609918836479251907448441, 18.82139231592738730095742288757, 20.04323432787257162339750362598, 21.10971648342580688461475303357, 22.32562449745124228996731704380, 22.910939379605145693483226413535, 23.835393442052900218563798399279, 25.19989996575153358665070925256, 26.623153078339937784207705027349, 27.24774965148339622682720114091, 28.114600248776143366659447294