L(s) = 1 | + (0.610 + 0.791i)2-s + (0.309 − 0.951i)3-s + (−0.254 + 0.967i)4-s + (0.941 − 0.336i)6-s + (−0.921 + 0.389i)8-s + (−0.809 − 0.587i)9-s + (0.841 + 0.540i)12-s + (0.362 + 0.931i)13-s + (−0.870 − 0.491i)16-s + (−0.696 − 0.717i)17-s + (−0.0285 − 0.999i)18-s + (0.897 + 0.441i)19-s + (−0.415 − 0.909i)23-s + (0.0855 + 0.996i)24-s + (−0.516 + 0.856i)26-s + (−0.809 + 0.587i)27-s + ⋯ |
L(s) = 1 | + (0.610 + 0.791i)2-s + (0.309 − 0.951i)3-s + (−0.254 + 0.967i)4-s + (0.941 − 0.336i)6-s + (−0.921 + 0.389i)8-s + (−0.809 − 0.587i)9-s + (0.841 + 0.540i)12-s + (0.362 + 0.931i)13-s + (−0.870 − 0.491i)16-s + (−0.696 − 0.717i)17-s + (−0.0285 − 0.999i)18-s + (0.897 + 0.441i)19-s + (−0.415 − 0.909i)23-s + (0.0855 + 0.996i)24-s + (−0.516 + 0.856i)26-s + (−0.809 + 0.587i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.366749376 + 0.1754507780i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.366749376 + 0.1754507780i\) |
\(L(1)\) |
\(\approx\) |
\(1.454692712 + 0.2416920079i\) |
\(L(1)\) |
\(\approx\) |
\(1.454692712 + 0.2416920079i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.610 + 0.791i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 13 | \( 1 + (0.362 + 0.931i)T \) |
| 17 | \( 1 + (-0.696 - 0.717i)T \) |
| 19 | \( 1 + (0.897 + 0.441i)T \) |
| 23 | \( 1 + (-0.415 - 0.909i)T \) |
| 29 | \( 1 + (0.985 - 0.170i)T \) |
| 31 | \( 1 + (0.998 + 0.0570i)T \) |
| 37 | \( 1 + (-0.774 - 0.633i)T \) |
| 41 | \( 1 + (-0.564 + 0.825i)T \) |
| 43 | \( 1 + (-0.654 - 0.755i)T \) |
| 47 | \( 1 + (-0.0285 + 0.999i)T \) |
| 53 | \( 1 + (0.870 - 0.491i)T \) |
| 59 | \( 1 + (0.564 + 0.825i)T \) |
| 61 | \( 1 + (0.610 - 0.791i)T \) |
| 67 | \( 1 + (0.959 + 0.281i)T \) |
| 71 | \( 1 + (-0.466 - 0.884i)T \) |
| 73 | \( 1 + (0.736 - 0.676i)T \) |
| 79 | \( 1 + (-0.974 + 0.226i)T \) |
| 83 | \( 1 + (-0.198 - 0.980i)T \) |
| 89 | \( 1 + (0.142 + 0.989i)T \) |
| 97 | \( 1 + (0.0855 + 0.996i)T \) |
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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
−18.45317427067698069650432755739, −17.686467947541027871428056814102, −17.12161330947806235205388872018, −15.79820612580953515994246668489, −15.66476595649738573404723602221, −15.01767475335982038034427130795, −14.169249334425897786209110853335, −13.59306863409079893621827739594, −13.092953758167358896453448756615, −12.06405528335018243946757654521, −11.483725138950258647095769226761, −10.82725828402732028098109469653, −9.9664070642228689908743836078, −9.91451748508773284128391574898, −8.610089569119615111291810271206, −8.43929253187854454970672390708, −7.10846736593164856824699274097, −6.13447485433892339441792706584, −5.39841406602024436645118573598, −4.87226792167261732292901128183, −4.00831274542036513405435518369, −3.39248828018231696527883534783, −2.78254552444327750459520926366, −1.89404736066740192234210815186, −0.80923093948501018607648785198,
0.64998536148852936210302273960, 1.87528284852974495449682218381, 2.652319783321132597523185620534, 3.41293976148318023934546800877, 4.29161060522257179050123302254, 5.05437835122121773845205731965, 5.94939440425382836718788586006, 6.66977585673824294766611572590, 6.97554963728970191064531120177, 7.90823187860662304844044225458, 8.49154395117215721408611224956, 9.06691239642129177791376199754, 9.967906670774867703777938810551, 11.224674716576027069626059566, 11.92549085680664890260765033302, 12.23419587362208528491331341736, 13.23323378984668498254742375427, 13.72925277445511856759630942409, 14.16644815970456797173885002202, 14.8112666469287700833433863104, 15.77623013736898071078405812332, 16.202678854418627044574415691792, 17.02815751527999575477938603114, 17.745611133011216691459620193662, 18.26788647653816733023415738936