L(s) = 1 | + (−0.559 − 0.829i)2-s + (−0.374 + 0.927i)4-s + (−0.766 − 0.642i)5-s + (0.0348 − 0.999i)7-s + (0.978 − 0.207i)8-s + (−0.104 + 0.994i)10-s + (−0.0348 + 0.999i)11-s + (0.438 + 0.898i)13-s + (−0.848 + 0.529i)14-s + (−0.719 − 0.694i)16-s + (−0.309 + 0.951i)17-s + (0.913 − 0.406i)19-s + (0.882 − 0.469i)20-s + (0.848 − 0.529i)22-s + (−0.848 + 0.529i)23-s + ⋯ |
L(s) = 1 | + (−0.559 − 0.829i)2-s + (−0.374 + 0.927i)4-s + (−0.766 − 0.642i)5-s + (0.0348 − 0.999i)7-s + (0.978 − 0.207i)8-s + (−0.104 + 0.994i)10-s + (−0.0348 + 0.999i)11-s + (0.438 + 0.898i)13-s + (−0.848 + 0.529i)14-s + (−0.719 − 0.694i)16-s + (−0.309 + 0.951i)17-s + (0.913 − 0.406i)19-s + (0.882 − 0.469i)20-s + (0.848 − 0.529i)22-s + (−0.848 + 0.529i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.07013863987 - 0.3701251721i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.07013863987 - 0.3701251721i\) |
\(L(1)\) |
\(\approx\) |
\(0.5657560469 - 0.2797466823i\) |
\(L(1)\) |
\(\approx\) |
\(0.5657560469 - 0.2797466823i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.559 - 0.829i)T \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (0.0348 - 0.999i)T \) |
| 11 | \( 1 + (-0.0348 + 0.999i)T \) |
| 13 | \( 1 + (0.438 + 0.898i)T \) |
| 17 | \( 1 + (-0.309 + 0.951i)T \) |
| 19 | \( 1 + (0.913 - 0.406i)T \) |
| 23 | \( 1 + (-0.848 + 0.529i)T \) |
| 29 | \( 1 + (-0.559 - 0.829i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.719 - 0.694i)T \) |
| 43 | \( 1 + (-0.997 + 0.0697i)T \) |
| 47 | \( 1 + (0.241 - 0.970i)T \) |
| 53 | \( 1 + (-0.669 - 0.743i)T \) |
| 59 | \( 1 + (-0.559 + 0.829i)T \) |
| 61 | \( 1 + (0.766 - 0.642i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + (0.978 - 0.207i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (0.990 + 0.139i)T \) |
| 83 | \( 1 + (0.997 - 0.0697i)T \) |
| 89 | \( 1 + (-0.669 + 0.743i)T \) |
| 97 | \( 1 + (-0.882 + 0.469i)T \) |
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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
−22.357506034674997284234697432419, −22.05821229376155876352612392636, −20.54423271526696305424782458047, −19.81267487123367517378531798657, −18.86575410967133551589893808678, −18.30148517588256703923539207725, −17.94978966299068695779882963233, −16.42254896614493736748297766109, −16.018479822583171513029559913264, −15.32801872347212781674842768115, −14.5191370680954502947355504889, −13.81887192062352237284936744237, −12.65175573008205009279182071780, −11.49456253521220492183154480263, −10.9692468535505276628631900664, −9.899142438494051571093009760509, −8.97907583552667895438012634159, −8.131709645616188558722750195989, −7.61450788814625423637176269402, −6.424547120001185274300939818681, −5.79456945014751938871065787567, −4.85791757160763869750022297039, −3.50413569561296858433781272766, −2.571579589083543729472180393761, −0.97243502327955702240547902705,
0.13039557683648222704618696153, 1.2253689418378453282851896479, 2.08580763361733451636082709894, 3.72657904166186010624262103001, 4.04952104854478921552241815425, 4.99168423003979165781716959816, 6.69771506253312742806221977092, 7.60523174934782735767959673193, 8.15679072869083319637794026732, 9.28727523270362766596224107653, 9.863194592301278046463739748894, 10.95848586566814966525800102630, 11.57252742469224922261382348258, 12.37663645030158265892895991369, 13.21036788177746815140202242912, 13.88996939589422709794341003906, 15.188068391110506387105762294097, 16.13111631361798600661128320042, 16.827318493620320370696255007957, 17.515663494738106080881127857255, 18.356728606992130737484075259095, 19.399217994722432347281177698426, 19.92014255756429335667474980025, 20.46734015559011130904061396763, 21.2177578719789432500375557088