L(s) = 1 | + (−0.424 − 0.905i)2-s + (0.207 − 0.978i)3-s + (−0.640 + 0.768i)4-s + (−0.974 + 0.226i)6-s + (0.967 + 0.254i)8-s + (−0.913 − 0.406i)9-s + (0.618 + 0.786i)12-s + (0.717 − 0.696i)13-s + (−0.179 − 0.983i)16-s + (0.999 + 0.00951i)17-s + (0.0190 + 0.999i)18-s + (0.953 + 0.299i)19-s + (−0.690 + 0.723i)23-s + (0.449 − 0.893i)24-s + (−0.935 − 0.353i)26-s + (−0.587 + 0.809i)27-s + ⋯ |
L(s) = 1 | + (−0.424 − 0.905i)2-s + (0.207 − 0.978i)3-s + (−0.640 + 0.768i)4-s + (−0.974 + 0.226i)6-s + (0.967 + 0.254i)8-s + (−0.913 − 0.406i)9-s + (0.618 + 0.786i)12-s + (0.717 − 0.696i)13-s + (−0.179 − 0.983i)16-s + (0.999 + 0.00951i)17-s + (0.0190 + 0.999i)18-s + (0.953 + 0.299i)19-s + (−0.690 + 0.723i)23-s + (0.449 − 0.893i)24-s + (−0.935 − 0.353i)26-s + (−0.587 + 0.809i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00830 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00830 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.081251068 - 1.072312120i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.081251068 - 1.072312120i\) |
\(L(1)\) |
\(\approx\) |
\(0.7677934696 - 0.5837849033i\) |
\(L(1)\) |
\(\approx\) |
\(0.7677934696 - 0.5837849033i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.424 - 0.905i)T \) |
| 3 | \( 1 + (0.207 - 0.978i)T \) |
| 13 | \( 1 + (0.717 - 0.696i)T \) |
| 17 | \( 1 + (0.999 + 0.00951i)T \) |
| 19 | \( 1 + (0.953 + 0.299i)T \) |
| 23 | \( 1 + (-0.690 + 0.723i)T \) |
| 29 | \( 1 + (0.993 - 0.113i)T \) |
| 31 | \( 1 + (0.999 + 0.0380i)T \) |
| 37 | \( 1 + (-0.997 + 0.0665i)T \) |
| 41 | \( 1 + (0.921 + 0.389i)T \) |
| 43 | \( 1 + (0.540 - 0.841i)T \) |
| 47 | \( 1 + (-0.0190 + 0.999i)T \) |
| 53 | \( 1 + (-0.983 - 0.179i)T \) |
| 59 | \( 1 + (-0.123 + 0.992i)T \) |
| 61 | \( 1 + (-0.820 + 0.572i)T \) |
| 67 | \( 1 + (-0.945 + 0.327i)T \) |
| 71 | \( 1 + (0.198 - 0.980i)T \) |
| 73 | \( 1 + (-0.524 + 0.851i)T \) |
| 79 | \( 1 + (0.988 - 0.151i)T \) |
| 83 | \( 1 + (0.791 - 0.610i)T \) |
| 89 | \( 1 + (0.995 - 0.0950i)T \) |
| 97 | \( 1 + (0.0570 + 0.998i)T \) |
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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
−18.40920056111379610506855037710, −17.73981669726440702085518246424, −17.017276855094412442937809200949, −16.33701658808564550115050374812, −15.90767413242113138910437090357, −15.43247323722152382733171160989, −14.41784713585516854081343942302, −14.11216778919428104795403543499, −13.56861409143675740557068545392, −12.362270899359291308801299636982, −11.55901085355049412989460137934, −10.69207543153122034478020279402, −10.15598680138034484063982878823, −9.46524960738188898133131643970, −8.90432901666612533571830605904, −8.17314865446692766443496584264, −7.6167866722176210661330572163, −6.52830367145939262581916553930, −6.00672985835279790897075016452, −5.12384870219353829441977520048, −4.56857801441198913679080119710, −3.76797413992256068237427883890, −2.94757478006863266335364921508, −1.74600453120613490720227847670, −0.66660068003995135694626458808,
0.87553995440394650943222595784, 1.27443565492100422538727633705, 2.247373485492993177835841168615, 3.16013619286913077085887641452, 3.469292389197577470008697984386, 4.65723402782038648188904623953, 5.64577442141265956813540741296, 6.26251068795094215768269866472, 7.56595659053685371943309138784, 7.690683336781089095449048286663, 8.5212575166261865943291915942, 9.19650676560515606672574081410, 10.03981277731402595514416596155, 10.63347798249766924796767953341, 11.56562543575500905403447032702, 12.08664352279415244162070155690, 12.5340205559865016334999495168, 13.469746642698193378952833693522, 13.82493904795536772653079265674, 14.47261149095656449928304965760, 15.64268348688833341575465705584, 16.28912608069728519601291385540, 17.25622369884806449528332325368, 17.77704432728591367365465669054, 18.20212779956163563370838004867