L(s) = 1 | + (0.939 − 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.173 + 0.984i)5-s + (−0.173 − 0.984i)6-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.173 + 0.984i)10-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.173 − 0.984i)14-s + (0.939 + 0.342i)15-s + (0.173 − 0.984i)16-s + (0.173 − 0.984i)17-s − 18-s + ⋯ |
L(s) = 1 | + (0.939 − 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.173 + 0.984i)5-s + (−0.173 − 0.984i)6-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.173 + 0.984i)10-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + (0.173 − 0.984i)14-s + (0.939 + 0.342i)15-s + (0.173 − 0.984i)16-s + (0.173 − 0.984i)17-s − 18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.676487328 - 1.482389288i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.676487328 - 1.482389288i\) |
\(L(1)\) |
\(\approx\) |
\(1.643640653 - 0.8740918415i\) |
\(L(1)\) |
\(\approx\) |
\(1.643640653 - 0.8740918415i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 3 | \( 1 + (0.173 - 0.984i)T \) |
| 5 | \( 1 + (-0.173 + 0.984i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.173 - 0.984i)T \) |
| 23 | \( 1 + (-0.939 - 0.342i)T \) |
| 29 | \( 1 + (0.173 + 0.984i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.173 + 0.984i)T \) |
| 43 | \( 1 + (-0.939 + 0.342i)T \) |
| 47 | \( 1 + (-0.766 + 0.642i)T \) |
| 53 | \( 1 + (0.173 + 0.984i)T \) |
| 59 | \( 1 + (-0.173 + 0.984i)T \) |
| 61 | \( 1 + (0.766 - 0.642i)T \) |
| 67 | \( 1 + (-0.173 - 0.984i)T \) |
| 71 | \( 1 + (0.939 - 0.342i)T \) |
| 73 | \( 1 + (0.939 - 0.342i)T \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.766 + 0.642i)T \) |
| 97 | \( 1 + (-0.173 + 0.984i)T \) |
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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
−26.127385309203690166731246660655, −25.18139916311832558249147367095, −24.40864737956008729082328442748, −23.66973257279472402355438929589, −22.38258143792465428076318348978, −21.64301001025876682073696867117, −21.08051916611721811485246179471, −20.1959645904002997615926951758, −19.223500220010343418723730583122, −17.36943036959957442484211384994, −16.69345343597937832374493588830, −15.77074949130148840460610934820, −15.12781646481748327932460210621, −14.15243896659147223219265758173, −13.16512834093592450626023327573, −11.86670778354038897021759679516, −11.431074993987573958000848673, −9.888906891547692609947858738293, −8.4820534090412017887344778770, −8.195812523871087487407627244190, −6.08675142031729202047901511594, −5.412159171556206293844351145341, −4.37281463047536497000741913, −3.52842450103621286270089722705, −2.040091704230972926929705378695,
1.36202781964140404840304973479, 2.51626409500325828253563744464, 3.6117314542549278587907814178, 4.85282912716529094074775156647, 6.38920304307745870003824957771, 7.03606740047424669305494193843, 7.84764533962447189446532658648, 9.75662438134534274508249043293, 10.86209815113183736371639996875, 11.701587602405390045540985249459, 12.52789307310152613168690292971, 13.786998980923332903682664475266, 14.25430136718388902255885247223, 15.007775840524646400724975156239, 16.41670639102687489003782317143, 17.80658843673831796899640834230, 18.45542001265426008018042802509, 19.83850749559835854070776012249, 19.9963430698426612432564362513, 21.29655448312792438672168728853, 22.545257066077828511962915253435, 23.08677228729481894990563927094, 23.78228555851954997672805585211, 24.82179985107018327086778286889, 25.57210836180795137626770711171