L(s) = 1 | + (0.0475 − 0.998i)2-s + (−0.5 + 0.866i)3-s + (−0.995 − 0.0950i)4-s + (−0.786 + 0.618i)5-s + (0.841 + 0.540i)6-s + (−0.142 + 0.989i)8-s + (−0.5 − 0.866i)9-s + (0.580 + 0.814i)10-s + (0.580 − 0.814i)12-s + (0.415 + 0.909i)13-s + (−0.142 − 0.989i)15-s + (0.981 + 0.189i)16-s + (0.235 − 0.971i)17-s + (−0.888 + 0.458i)18-s + (0.235 + 0.971i)19-s + (0.841 − 0.540i)20-s + ⋯ |
L(s) = 1 | + (0.0475 − 0.998i)2-s + (−0.5 + 0.866i)3-s + (−0.995 − 0.0950i)4-s + (−0.786 + 0.618i)5-s + (0.841 + 0.540i)6-s + (−0.142 + 0.989i)8-s + (−0.5 − 0.866i)9-s + (0.580 + 0.814i)10-s + (0.580 − 0.814i)12-s + (0.415 + 0.909i)13-s + (−0.142 − 0.989i)15-s + (0.981 + 0.189i)16-s + (0.235 − 0.971i)17-s + (−0.888 + 0.458i)18-s + (0.235 + 0.971i)19-s + (0.841 − 0.540i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3001480201 + 0.4246909767i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3001480201 + 0.4246909767i\) |
\(L(1)\) |
\(\approx\) |
\(0.6569176385 + 0.02520116390i\) |
\(L(1)\) |
\(\approx\) |
\(0.6569176385 + 0.02520116390i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.0475 - 0.998i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.786 + 0.618i)T \) |
| 13 | \( 1 + (0.415 + 0.909i)T \) |
| 17 | \( 1 + (0.235 - 0.971i)T \) |
| 19 | \( 1 + (0.235 + 0.971i)T \) |
| 23 | \( 1 + (0.981 + 0.189i)T \) |
| 29 | \( 1 + (-0.959 - 0.281i)T \) |
| 31 | \( 1 + (0.580 + 0.814i)T \) |
| 37 | \( 1 + (-0.995 + 0.0950i)T \) |
| 41 | \( 1 + (0.841 + 0.540i)T \) |
| 43 | \( 1 + (-0.142 + 0.989i)T \) |
| 47 | \( 1 + (-0.888 - 0.458i)T \) |
| 53 | \( 1 + (0.981 - 0.189i)T \) |
| 59 | \( 1 + (0.0475 + 0.998i)T \) |
| 61 | \( 1 + (-0.888 - 0.458i)T \) |
| 67 | \( 1 + (-0.888 + 0.458i)T \) |
| 71 | \( 1 + (-0.959 - 0.281i)T \) |
| 73 | \( 1 + (-0.327 - 0.945i)T \) |
| 79 | \( 1 + (-0.786 + 0.618i)T \) |
| 83 | \( 1 + (-0.654 - 0.755i)T \) |
| 89 | \( 1 + (0.723 + 0.690i)T \) |
| 97 | \( 1 + (-0.142 + 0.989i)T \) |
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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
−22.3812569230786274502220515969, −21.16578106248998289735072683094, −20.06618667412070027186809160940, −19.197299766596304808973808533870, −18.65327993019612026252957955938, −17.56657437344951877644906008736, −17.15736762907709472854797449880, −16.31358669477821472010000341102, −15.49036640575719598540156858736, −14.83148773962874606797651646656, −13.59450948241017058375207861007, −12.97093045648443329024095983370, −12.43755871561315624168984754928, −11.38187903991793497421312425192, −10.46480486783971928157569033374, −9.025586194255743141200363482593, −8.38090568753433702151528407619, −7.59245303283145635624931401479, −6.96401486171194438860294249736, −5.85228404118734854815556619716, −5.24432776589860395255010853292, −4.25202005485664938273259762366, −3.11809360014225626075594370120, −1.33168802190806493755863346674, −0.29947661537361698398988596973,
1.26833861536455112643575193578, 2.84034590561480220790765067369, 3.53746097785779186672023625492, 4.333446390555032952745848252977, 5.15458704414652252750792622975, 6.26589850688627021370656620918, 7.4359253183058691750162972004, 8.62516954134859183286248995480, 9.4046842711921454468957867334, 10.25189435551390525338174879116, 10.98546596593648081501994765398, 11.69434716698290144437161010146, 12.09325133789368978095040735299, 13.40699044093893214623459959986, 14.39208929882443080169268869545, 14.931377787420706929371268480781, 16.053827485937753108285783909596, 16.65159482541905327179561961663, 17.784561599034982589156018466236, 18.5051922512607522175325071341, 19.1981307517954990625948757609, 20.06178034535370803415495924802, 21.00997937120222181431863613991, 21.31312238177783111058097321168, 22.44050024536591243493367629577