L(s) = 1 | + (−0.913 + 0.406i)2-s + (0.978 − 0.207i)3-s + (0.669 − 0.743i)4-s + (0.104 + 0.994i)5-s + (−0.809 + 0.587i)6-s + (−0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)15-s + (−0.104 − 0.994i)16-s + (0.913 + 0.406i)17-s + (−0.669 + 0.743i)18-s + (0.669 + 0.743i)19-s + (0.809 + 0.587i)20-s + ⋯ |
L(s) = 1 | + (−0.913 + 0.406i)2-s + (0.978 − 0.207i)3-s + (0.669 − 0.743i)4-s + (0.104 + 0.994i)5-s + (−0.809 + 0.587i)6-s + (−0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)15-s + (−0.104 − 0.994i)16-s + (0.913 + 0.406i)17-s + (−0.669 + 0.743i)18-s + (0.669 + 0.743i)19-s + (0.809 + 0.587i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8406974657 + 0.2731800534i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8406974657 + 0.2731800534i\) |
\(L(1)\) |
\(\approx\) |
\(0.9196500077 + 0.2031795045i\) |
\(L(1)\) |
\(\approx\) |
\(0.9196500077 + 0.2031795045i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.913 + 0.406i)T \) |
| 3 | \( 1 + (0.978 - 0.207i)T \) |
| 5 | \( 1 + (0.104 + 0.994i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (0.913 + 0.406i)T \) |
| 19 | \( 1 + (0.669 + 0.743i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (0.104 - 0.994i)T \) |
| 37 | \( 1 + (-0.978 - 0.207i)T \) |
| 41 | \( 1 + (0.309 - 0.951i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.669 - 0.743i)T \) |
| 53 | \( 1 + (-0.104 + 0.994i)T \) |
| 59 | \( 1 + (-0.669 + 0.743i)T \) |
| 61 | \( 1 + (-0.104 - 0.994i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (0.669 - 0.743i)T \) |
| 79 | \( 1 + (-0.913 + 0.406i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.809 + 0.587i)T \) |
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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
−31.14610950933427270280139428596, −29.99556186087838851066682182567, −28.88054488668012406940976609311, −27.84994463704120146471713226799, −26.90629498121278440740632075505, −25.93546602287636432280279888849, −24.90847211099217812601186257754, −24.14049207324646902297422748052, −21.90866372596963904032967711848, −20.97184042008757408406799810869, −20.11846319611519512013006710524, −19.32840881927045792368139483676, −18.07460673314366034789954296998, −16.66251204640140288025738924480, −15.91869609537698042275553811226, −14.34574584973031087057403572868, −12.93197091301434679647108243664, −11.8818612460793777106010364304, −10.11029646380673856669776455690, −9.25519487906222818225135745019, −8.31248289487468820425943493927, −7.12805130420861605672335124611, −4.74424697697348573701475329016, −3.09408260600771876590168807003, −1.57992341387840411782284677573,
1.94405889788413251251938742741, 3.31978586773099274497471833575, 5.8467012543475967381524582181, 7.33056613064629581835877621719, 7.964396457364982736691726541263, 9.602622245602518176958761337050, 10.29044780998396339784303488360, 11.958805292368458824441793809493, 13.82990819795933214564538990761, 14.77724507698132911452154167000, 15.56957565473373073805526002553, 17.18639954902215638532803296005, 18.36149227162872630782726914371, 19.10601188693525077477025064740, 20.07112980659551774255286739013, 21.30038121099317273673238686765, 22.86913129359824596942470538450, 24.25372643721155528372721149299, 25.1982922054422131636767483274, 26.05755090325791665068397747848, 26.81918689190506805679285756565, 27.7898568519067534282835368003, 29.519456756138877336294925539077, 29.9902286000681655484547324801, 31.4036645624177693313923299585