L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − i·7-s + i·8-s − 9-s + i·12-s + i·13-s − 14-s + 16-s − i·17-s + i·18-s + 19-s − 21-s + ⋯ |
L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − i·7-s + i·8-s − 9-s + i·12-s + i·13-s − 14-s + 16-s − i·17-s + i·18-s + 19-s − 21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2095296766 - 0.7375752672i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2095296766 - 0.7375752672i\) |
\(L(1)\) |
\(\approx\) |
\(0.5681825932 - 0.6908970148i\) |
\(L(1)\) |
\(\approx\) |
\(0.5681825932 - 0.6908970148i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 37 | \( 1 \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - iT \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + T \) |
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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
−33.55974636240407206389437111767, −32.549007131244852016923870742324, −31.78150812078579489952822386590, −30.6956928537032805792928413518, −28.57715824334067897918165744472, −27.73930261508734232438839379529, −26.75242517794643683342436564489, −25.6249728337038806296669379924, −24.769621705110224692389199204, −23.24629128720234115597461017040, −22.18706704960063304768598865595, −21.36936236511018472333035383780, −19.670082160581298652304251505741, −18.11319170454102123632588394468, −17.05887107232046072157090689955, −15.63748650676019008009904308135, −15.232324529844928647522678750276, −13.822996443558026762192561296946, −12.16537747871155698957330730347, −10.31416767897367231854698175471, −9.1138940445649639357805213867, −8.0202951658370410192258206025, −5.99327372809511249394729296399, −5.05253762399085146339272978849, −3.36821907198822085139335372618,
1.17155576578275287654596552465, 2.8983068119797527576009841846, 4.692444821709162981566826113627, 6.7426346968167754797254702577, 8.1835189412554803394645940004, 9.7198889901411094722380200637, 11.23110802001590907634234554970, 12.19129976380949201954467889363, 13.601241601754318067842414513151, 14.14459172543860594237408538551, 16.64768212483750881771031802450, 17.84589802006902908395438349915, 18.848247921416292540248810854615, 19.913957582672861860996791713581, 20.81606794903912392965766668972, 22.47438114365105007497242594099, 23.35837913542121270306685369852, 24.44524607677033870636521768993, 26.11274123694921689375997661899, 27.074604754124605231722921176582, 28.694028133055617448315206884096, 29.267205040598442276655835442490, 30.40228607668609348983391073773, 31.082281997136659420455691185834, 32.36846222489207584606431003901