L(s) = 1 | + (0.406 + 0.913i)2-s + (0.965 + 0.258i)3-s + (−0.669 + 0.743i)4-s + (0.207 − 0.978i)5-s + (0.156 + 0.987i)6-s + (−0.951 − 0.309i)8-s + (0.866 + 0.5i)9-s + (0.978 − 0.207i)10-s + (0.544 − 0.838i)11-s + (−0.838 + 0.544i)12-s + (0.987 − 0.156i)13-s + (0.453 − 0.891i)15-s + (−0.104 − 0.994i)16-s + (0.838 + 0.544i)17-s + (−0.104 + 0.994i)18-s + (−0.629 + 0.777i)19-s + ⋯ |
L(s) = 1 | + (0.406 + 0.913i)2-s + (0.965 + 0.258i)3-s + (−0.669 + 0.743i)4-s + (0.207 − 0.978i)5-s + (0.156 + 0.987i)6-s + (−0.951 − 0.309i)8-s + (0.866 + 0.5i)9-s + (0.978 − 0.207i)10-s + (0.544 − 0.838i)11-s + (−0.838 + 0.544i)12-s + (0.987 − 0.156i)13-s + (0.453 − 0.891i)15-s + (−0.104 − 0.994i)16-s + (0.838 + 0.544i)17-s + (−0.104 + 0.994i)18-s + (−0.629 + 0.777i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.786476170 + 1.095914915i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.786476170 + 1.095914915i\) |
\(L(1)\) |
\(\approx\) |
\(1.546495544 + 0.7136813194i\) |
\(L(1)\) |
\(\approx\) |
\(1.546495544 + 0.7136813194i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.406 + 0.913i)T \) |
| 3 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 + (0.207 - 0.978i)T \) |
| 11 | \( 1 + (0.544 - 0.838i)T \) |
| 13 | \( 1 + (0.987 - 0.156i)T \) |
| 17 | \( 1 + (0.838 + 0.544i)T \) |
| 19 | \( 1 + (-0.629 + 0.777i)T \) |
| 23 | \( 1 + (-0.913 + 0.406i)T \) |
| 29 | \( 1 + (0.891 + 0.453i)T \) |
| 31 | \( 1 + (-0.978 + 0.207i)T \) |
| 37 | \( 1 + (-0.978 - 0.207i)T \) |
| 43 | \( 1 + (0.587 - 0.809i)T \) |
| 47 | \( 1 + (-0.933 - 0.358i)T \) |
| 53 | \( 1 + (0.0523 + 0.998i)T \) |
| 59 | \( 1 + (0.104 - 0.994i)T \) |
| 61 | \( 1 + (-0.994 + 0.104i)T \) |
| 67 | \( 1 + (0.998 - 0.0523i)T \) |
| 71 | \( 1 + (-0.453 - 0.891i)T \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (-0.258 - 0.965i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.777 + 0.629i)T \) |
| 97 | \( 1 + (-0.453 + 0.891i)T \) |
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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
−25.63616390603130784809030259116, −24.40397945238278254636195639901, −23.32777303899757868014728069502, −22.622457289641512361404301888845, −21.55452289610369700262140926053, −20.861806753916041469223120227170, −19.926123033391093644182031745976, −19.170084548990950540977336188, −18.35126357965335235205577207717, −17.69291551093108744145385235906, −15.77036512667044775974738520638, −14.76360251035201532105291186311, −14.206410242171120319112047078864, −13.39486050141176786793205998086, −12.37927319016292330898026114165, −11.35469325285871253222137572738, −10.22848610069532786828323306265, −9.52241810628170001232835822632, −8.43472212928120932295377100559, −7.06938482428644248353580142381, −6.10384841236270739760381867518, −4.413690172581413954472679568351, −3.4767024487326460137573089805, −2.510715429722892274945560437792, −1.53165474716435098994771381162,
1.52889220512988031296907661232, 3.44315130812804988289671232005, 4.0466591253272215557775791234, 5.37388895833170628437298411273, 6.28185192892991813727453394758, 7.80587568075219969113699031265, 8.5103747061347208168799729222, 9.11079497849984828197143504163, 10.356915851918671049566584177853, 12.12240203783994884594961925885, 12.97090492737357433598615780170, 13.88317039386260547238672879320, 14.437795251828910688741956066934, 15.704036717895873110152564431916, 16.27472743234623719331283308664, 17.10479288154800656502427979703, 18.356963028222801669439596990150, 19.36025267428451679141772711720, 20.48419126730470370473298997628, 21.29908900427464010071334436333, 21.85732328873070874917120344168, 23.30840945008583539134624330335, 24.04130034779206697693090628154, 24.92582688360301401927287124136, 25.50576512282694991977722334906