L(s) = 1 | + (−0.774 − 0.632i)2-s + (−0.428 − 0.903i)3-s + (0.200 + 0.979i)4-s + (−0.600 − 0.799i)5-s + (−0.239 + 0.970i)6-s + (0.464 − 0.885i)8-s + (−0.632 + 0.774i)9-s + (−0.0402 + 0.999i)10-s + (−0.774 + 0.632i)11-s + (0.799 − 0.600i)12-s + (−0.464 + 0.885i)15-s + (−0.919 + 0.391i)16-s + (−0.845 + 0.534i)17-s + (0.979 − 0.200i)18-s + (−0.866 + 0.5i)19-s + (0.663 − 0.748i)20-s + ⋯ |
L(s) = 1 | + (−0.774 − 0.632i)2-s + (−0.428 − 0.903i)3-s + (0.200 + 0.979i)4-s + (−0.600 − 0.799i)5-s + (−0.239 + 0.970i)6-s + (0.464 − 0.885i)8-s + (−0.632 + 0.774i)9-s + (−0.0402 + 0.999i)10-s + (−0.774 + 0.632i)11-s + (0.799 − 0.600i)12-s + (−0.464 + 0.885i)15-s + (−0.919 + 0.391i)16-s + (−0.845 + 0.534i)17-s + (0.979 − 0.200i)18-s + (−0.866 + 0.5i)19-s + (0.663 − 0.748i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0250 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0250 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3213535449 - 0.3133989621i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3213535449 - 0.3133989621i\) |
\(L(1)\) |
\(\approx\) |
\(0.4121246539 - 0.2520672871i\) |
\(L(1)\) |
\(\approx\) |
\(0.4121246539 - 0.2520672871i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.774 - 0.632i)T \) |
| 3 | \( 1 + (-0.428 - 0.903i)T \) |
| 5 | \( 1 + (-0.600 - 0.799i)T \) |
| 11 | \( 1 + (-0.774 + 0.632i)T \) |
| 17 | \( 1 + (-0.845 + 0.534i)T \) |
| 19 | \( 1 + (-0.866 + 0.5i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.354 - 0.935i)T \) |
| 31 | \( 1 + (-0.721 - 0.692i)T \) |
| 37 | \( 1 + (0.960 - 0.278i)T \) |
| 41 | \( 1 + (-0.822 - 0.568i)T \) |
| 43 | \( 1 + (0.970 - 0.239i)T \) |
| 47 | \( 1 + (-0.979 - 0.200i)T \) |
| 53 | \( 1 + (-0.845 + 0.534i)T \) |
| 59 | \( 1 + (0.391 - 0.919i)T \) |
| 61 | \( 1 + (0.845 + 0.534i)T \) |
| 67 | \( 1 + (0.316 + 0.948i)T \) |
| 71 | \( 1 + (0.822 + 0.568i)T \) |
| 73 | \( 1 + (0.774 - 0.632i)T \) |
| 79 | \( 1 + (-0.200 + 0.979i)T \) |
| 83 | \( 1 + (0.822 - 0.568i)T \) |
| 89 | \( 1 + (0.866 - 0.5i)T \) |
| 97 | \( 1 + (-0.992 - 0.120i)T \) |
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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
−21.50590839338289696293939088172, −20.48015634420397082719421904526, −19.80656041323948843735626491558, −18.92490154569587220034584302822, −18.1931037030732833738037829070, −17.642695455914075856727876409114, −16.484973940049478189547506482866, −16.18631828503074174240946052193, −15.29665490032327118949051175404, −14.84879449945824951412784017602, −14.04238805248355858323000773031, −12.81887955152362021714422709770, −11.45179033101367022470811909755, −10.931177453300342968475946487988, −10.543883099274635755504384478353, −9.49683542727948344984996830553, −8.70042294465485842400262559818, −7.96493265655741324485222515006, −6.81358234720416585835468985135, −6.367732477223736184638789738765, −5.20544628830712500313858390946, −4.54705389002348879056435319341, −3.292174859559008272182454822888, −2.38253701028494090820402369405, −0.52221967820026755719320727267,
0.50971074236543443294518314298, 1.73285979307020596878583038099, 2.32931631496204486362062899474, 3.72974672714997498102693325813, 4.62576563265565965465876059860, 5.71857671554811883692372458186, 6.85937396896247748498409328414, 7.69014005271848449885498990317, 8.17189961195830368223105236940, 9.02827158261083648688597763515, 10.00484556049895840759329680391, 11.09080792389905091754297392849, 11.45554964631147223327610244893, 12.587877957839309517541096578250, 12.799308521180104304972779770592, 13.517378028579452762734058488372, 15.059394246260396773204531172299, 15.84765257279493161973722735810, 16.765544947768866290194870880814, 17.3067338363076621364382021428, 17.95105838268998096815715626261, 18.925955810498988761747693859076, 19.29914801504223024100847944803, 20.1783433539622589758982586524, 20.71779381879114873783511846512