L(s) = 1 | + (0.357 + 0.933i)2-s + (0.772 + 0.635i)3-s + (−0.744 + 0.668i)4-s + (0.436 + 0.899i)5-s + (−0.317 + 0.948i)6-s + (0.584 − 0.811i)7-s + (−0.890 − 0.455i)8-s + (0.192 + 0.981i)9-s + (−0.683 + 0.729i)10-s + (0.941 + 0.337i)11-s + (−0.999 + 0.0430i)12-s + (−0.798 − 0.601i)13-s + (0.966 + 0.255i)14-s + (−0.234 + 0.972i)15-s + (0.107 − 0.994i)16-s + (0.357 + 0.933i)17-s + ⋯ |
L(s) = 1 | + (0.357 + 0.933i)2-s + (0.772 + 0.635i)3-s + (−0.744 + 0.668i)4-s + (0.436 + 0.899i)5-s + (−0.317 + 0.948i)6-s + (0.584 − 0.811i)7-s + (−0.890 − 0.455i)8-s + (0.192 + 0.981i)9-s + (−0.683 + 0.729i)10-s + (0.941 + 0.337i)11-s + (−0.999 + 0.0430i)12-s + (−0.798 − 0.601i)13-s + (0.966 + 0.255i)14-s + (−0.234 + 0.972i)15-s + (0.107 − 0.994i)16-s + (0.357 + 0.933i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6776413116 + 1.848329798i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6776413116 + 1.848329798i\) |
\(L(1)\) |
\(\approx\) |
\(1.084608273 + 1.192051230i\) |
\(L(1)\) |
\(\approx\) |
\(1.084608273 + 1.192051230i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 293 | \( 1 \) |
good | 2 | \( 1 + (0.357 + 0.933i)T \) |
| 3 | \( 1 + (0.772 + 0.635i)T \) |
| 5 | \( 1 + (0.436 + 0.899i)T \) |
| 7 | \( 1 + (0.584 - 0.811i)T \) |
| 11 | \( 1 + (0.941 + 0.337i)T \) |
| 13 | \( 1 + (-0.798 - 0.601i)T \) |
| 17 | \( 1 + (0.357 + 0.933i)T \) |
| 19 | \( 1 + (-0.234 + 0.972i)T \) |
| 23 | \( 1 + (-0.317 - 0.948i)T \) |
| 29 | \( 1 + (-0.0645 - 0.997i)T \) |
| 31 | \( 1 + (0.436 - 0.899i)T \) |
| 37 | \( 1 + (-0.976 - 0.213i)T \) |
| 41 | \( 1 + (-0.954 + 0.296i)T \) |
| 43 | \( 1 + (-0.150 - 0.988i)T \) |
| 47 | \( 1 + (0.0215 - 0.999i)T \) |
| 53 | \( 1 + (0.941 + 0.337i)T \) |
| 59 | \( 1 + (0.996 + 0.0859i)T \) |
| 61 | \( 1 + (-0.397 + 0.917i)T \) |
| 67 | \( 1 + (0.985 - 0.171i)T \) |
| 71 | \( 1 + (0.512 - 0.858i)T \) |
| 73 | \( 1 + (-0.744 - 0.668i)T \) |
| 79 | \( 1 + (-0.397 + 0.917i)T \) |
| 83 | \( 1 + (-0.474 + 0.880i)T \) |
| 89 | \( 1 + (-0.397 - 0.917i)T \) |
| 97 | \( 1 + (0.869 + 0.493i)T \) |
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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
−24.74890537767390306128176900509, −24.354722208298943749843408313614, −23.48716811089773679074131959159, −21.954868551315680722260479805978, −21.46398786156350924709336050651, −20.54444526264371592221852611357, −19.72917091906983122993073130467, −19.07158713711159472106944877081, −17.99678400217013863872257509298, −17.28729639668543387961271020725, −15.68375840087244701490010179332, −14.4262347544795107201988891938, −14.00157408115078275452556056196, −12.95709120540729987712678883667, −12.04985462674555526159425507045, −11.572451603157518282263306671726, −9.73578988832636507604961892811, −9.07586633508531547054201847432, −8.44669939254189506184858966610, −6.83499923478896947622405519647, −5.46197206852734680101008766067, −4.57038492738756661853803700903, −3.15772327141368107723543999764, −2.044776414025365387562838532018, −1.22668015480783261437760352912,
2.13330091958099989652576585046, 3.59721234905932496481599154354, 4.235610487150590540415735139524, 5.534046255035290222261336048290, 6.75493974463767399048580202489, 7.68829202200046446632671239972, 8.49928956745241866167531968757, 9.91028444900485868408779297960, 10.36322069038506949720907314838, 11.98122890007172535983389682846, 13.34449578619935630803123668764, 14.1834303962409614436442846732, 14.75163603849643634905588000454, 15.274892490414324053821876110336, 16.90364895571109608429894592987, 17.11586101168973289961642387938, 18.42710892434279685165584216550, 19.46988596700806284260529109418, 20.62463128432535241560875129295, 21.440577580586128511919265349435, 22.38218512789357616897438927165, 22.8816586685064616553087119910, 24.26146109889846025387222369592, 25.014239657199063717854352069132, 25.771884614971354527780921035003