L(s) = 1 | + (−0.0402 − 0.999i)2-s + (0.919 − 0.391i)3-s + (−0.996 + 0.0804i)4-s + (0.987 + 0.160i)5-s + (−0.428 − 0.903i)6-s + (−0.799 − 0.600i)7-s + (0.120 + 0.992i)8-s + (0.692 − 0.721i)9-s + (0.120 − 0.992i)10-s + (−0.632 − 0.774i)11-s + (−0.885 + 0.464i)12-s + (0.428 − 0.903i)13-s + (−0.568 + 0.822i)14-s + (0.970 − 0.239i)15-s + (0.987 − 0.160i)16-s + (−0.568 − 0.822i)17-s + ⋯ |
L(s) = 1 | + (−0.0402 − 0.999i)2-s + (0.919 − 0.391i)3-s + (−0.996 + 0.0804i)4-s + (0.987 + 0.160i)5-s + (−0.428 − 0.903i)6-s + (−0.799 − 0.600i)7-s + (0.120 + 0.992i)8-s + (0.692 − 0.721i)9-s + (0.120 − 0.992i)10-s + (−0.632 − 0.774i)11-s + (−0.885 + 0.464i)12-s + (0.428 − 0.903i)13-s + (−0.568 + 0.822i)14-s + (0.970 − 0.239i)15-s + (0.987 − 0.160i)16-s + (−0.568 − 0.822i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6853233399 - 1.894885574i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6853233399 - 1.894885574i\) |
\(L(1)\) |
\(\approx\) |
\(0.9886460003 - 0.9666116484i\) |
\(L(1)\) |
\(\approx\) |
\(0.9886460003 - 0.9666116484i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 79 | \( 1 \) |
good | 2 | \( 1 + (-0.0402 - 0.999i)T \) |
| 3 | \( 1 + (0.919 - 0.391i)T \) |
| 5 | \( 1 + (0.987 + 0.160i)T \) |
| 7 | \( 1 + (-0.799 - 0.600i)T \) |
| 11 | \( 1 + (-0.632 - 0.774i)T \) |
| 13 | \( 1 + (0.428 - 0.903i)T \) |
| 17 | \( 1 + (-0.568 - 0.822i)T \) |
| 19 | \( 1 + (0.948 - 0.316i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.278 + 0.960i)T \) |
| 31 | \( 1 + (-0.845 + 0.534i)T \) |
| 37 | \( 1 + (0.200 - 0.979i)T \) |
| 41 | \( 1 + (0.354 + 0.935i)T \) |
| 43 | \( 1 + (0.632 - 0.774i)T \) |
| 47 | \( 1 + (0.200 + 0.979i)T \) |
| 53 | \( 1 + (0.919 + 0.391i)T \) |
| 59 | \( 1 + (0.996 + 0.0804i)T \) |
| 61 | \( 1 + (0.748 + 0.663i)T \) |
| 67 | \( 1 + (0.885 - 0.464i)T \) |
| 71 | \( 1 + (-0.120 - 0.992i)T \) |
| 73 | \( 1 + (0.428 + 0.903i)T \) |
| 83 | \( 1 + (-0.996 + 0.0804i)T \) |
| 89 | \( 1 + (0.120 - 0.992i)T \) |
| 97 | \( 1 + (-0.748 - 0.663i)T \) |
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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
−31.46473801141163783011976218645, −30.74436407161649183948846545108, −28.83584896544795430700802756688, −28.040860135020550485237718899294, −26.25619540797083622269893079813, −26.0766942741211153751482240874, −25.05444730512915142546339887410, −24.16627781823823991690128859592, −22.51915220136711055612983528541, −21.68910825291827557150066830098, −20.5187281687192293437103281975, −18.97979179895665618367665891766, −18.10713905442003105665913517734, −16.638567031981829814665782792170, −15.72034664384080317581273277728, −14.679525919600931088914829977580, −13.572801290683409362283890485843, −12.77547553349603959156260523615, −10.08520854909369121211771785280, −9.39034524741908150010911049610, −8.359861515614223976894884128210, −6.80945015744712713232044722289, −5.54263066036840103134521821177, −4.06680500102672849920694493409, −2.20550769778592720697942689649,
0.94233912607517186640046904553, 2.6200945908626861382630837774, 3.50389633598728632477832691052, 5.57794743191500333305160159495, 7.380882901438679668669969249635, 8.92354462629879402020250738745, 9.80859999141005567690627703863, 10.90991562661176897992738957769, 12.80372144992506270555992215071, 13.45397635265023830092418878517, 14.17115135073925207556620274937, 16.00944774067622345604047579878, 17.81023498866651242622722373228, 18.436974453575540548358930523157, 19.75616711744365494487930971021, 20.44564112230161728691944872506, 21.53476902881195380999281579363, 22.595614849193329138318806922189, 23.94889058723322442019658807613, 25.3901519671595596508067892510, 26.224200980390483475074126021206, 27.09584799302363765062630216504, 28.81954348249945615701825928721, 29.500919613985075302852930231786, 30.18457629310940447203891877829