L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.978 − 0.207i)5-s + (−0.913 − 0.406i)7-s + (−0.809 − 0.587i)8-s + (0.5 + 0.866i)10-s + (0.104 − 0.994i)14-s + (0.309 − 0.951i)16-s + (0.669 + 0.743i)17-s + (−0.913 + 0.406i)19-s + (−0.669 + 0.743i)20-s + (0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (0.978 − 0.207i)28-s + (−0.809 + 0.587i)29-s + ⋯ |
L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.978 − 0.207i)5-s + (−0.913 − 0.406i)7-s + (−0.809 − 0.587i)8-s + (0.5 + 0.866i)10-s + (0.104 − 0.994i)14-s + (0.309 − 0.951i)16-s + (0.669 + 0.743i)17-s + (−0.913 + 0.406i)19-s + (−0.669 + 0.743i)20-s + (0.5 − 0.866i)23-s + (0.913 − 0.406i)25-s + (0.978 − 0.207i)28-s + (−0.809 + 0.587i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.414822873 + 0.9660071617i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.414822873 + 0.9660071617i\) |
\(L(1)\) |
\(\approx\) |
\(1.107192519 + 0.5335953546i\) |
\(L(1)\) |
\(\approx\) |
\(1.107192519 + 0.5335953546i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (0.978 - 0.207i)T \) |
| 7 | \( 1 + (-0.913 - 0.406i)T \) |
| 17 | \( 1 + (0.669 + 0.743i)T \) |
| 19 | \( 1 + (-0.913 + 0.406i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (0.669 - 0.743i)T \) |
| 37 | \( 1 + (0.913 + 0.406i)T \) |
| 41 | \( 1 + (0.913 - 0.406i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.913 + 0.406i)T \) |
| 53 | \( 1 + (-0.309 - 0.951i)T \) |
| 59 | \( 1 + (0.809 - 0.587i)T \) |
| 61 | \( 1 + (0.978 - 0.207i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.669 - 0.743i)T \) |
| 73 | \( 1 + (0.809 - 0.587i)T \) |
| 79 | \( 1 + (0.978 + 0.207i)T \) |
| 83 | \( 1 + (0.669 + 0.743i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.669 - 0.743i)T \) |
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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
−21.13301786876370593968386780187, −20.142235638959391593758653001550, −19.268428167328707847559596608244, −18.81580739685379839300205039237, −17.96185263380288251814929634738, −17.283126588792383645487308930, −16.33506190868893837866666617341, −15.26668518217704434301593495448, −14.55392328260856729332704082456, −13.65570527218115788908979530363, −13.13031199358642254381085800844, −12.45706909730662748463597492866, −11.538698297135948558453619899805, −10.71131582153114681681898115611, −9.85300201937835221310093665843, −9.41265129404610800299827633164, −8.6621936130176838937159914525, −7.23995532451354895772008297251, −6.18024535597373976243701549283, −5.62937511287641488672964914580, −4.708001833819504990787052469461, −3.53414986466926406524164487750, −2.75648985532964952754280036187, −2.07148329116097669955376915000, −0.84123092544539045118522777778,
0.87387205642007534676503276827, 2.36134311189604252725226208885, 3.42742424931471672988139164376, 4.306813042761786556465324764714, 5.25942848163178958150996030960, 6.287361309470552044726930789716, 6.412656411112272083533804775160, 7.63313102213522637106485673564, 8.46617547356719662834791029600, 9.38394696335133131021665033727, 9.94009661086380398584326743480, 10.846093315115544462391029771269, 12.32241086162508214274130964668, 12.99488659178681660466854684746, 13.28919180489920192973377928641, 14.52283337437924708181367259394, 14.711079594923990018008302868912, 15.99991034515304815162360719580, 16.634636723045478830191192567604, 17.065960379845126162728432524383, 17.8765985120193537882997609494, 18.78073879667379994864597574515, 19.42260036038432393736319927297, 20.78828565316082466815418616290, 21.11907480208527725783715646288