L(s) = 1 | − i·7-s + i·11-s + 13-s + i·17-s − i·19-s − i·23-s − i·29-s − 31-s + 37-s + 41-s + 43-s − i·47-s − 49-s + 53-s − i·59-s + ⋯ |
L(s) = 1 | − i·7-s + i·11-s + 13-s + i·17-s − i·19-s − i·23-s − i·29-s − 31-s + 37-s + 41-s + 43-s − i·47-s − 49-s + 53-s − i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.635299016 - 0.8371394824i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.635299016 - 0.8371394824i\) |
\(L(1)\) |
\(\approx\) |
\(1.132334851 - 0.1837526503i\) |
\(L(1)\) |
\(\approx\) |
\(1.132334851 - 0.1837526503i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
| 31 | \( 1 \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 \) |
| 53 | \( 1 \) |
| 59 | \( 1 \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 \) |
| 83 | \( 1 \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 \) |
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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
−25.91551865155845505465711863110, −25.220473921467708583247442241134, −24.33582004063397806584274799811, −23.348265256082797725584061520253, −22.3456293114110479096347616038, −21.459615721221380374144874072957, −20.70832683954966057684908459433, −19.46544240469290585812154536853, −18.53415738638468113379032928286, −17.97467583438726641739159812349, −16.40320286901007101012210248880, −15.94265373662906456118329303325, −14.7196097634218867483583651545, −13.776835034792682778489168165731, −12.73215287712551204132785147624, −11.63879636835045480775313580456, −10.87606663623303646444782334960, −9.4127945155040995748423764259, −8.67287561409507563396018740962, −7.55083718207954576487158874209, −6.06330952045021560882620802710, −5.44054623464780956599534578108, −3.80516048151390820815877562248, −2.7062878592133200723299586384, −1.17435624206612555503805095663,
0.691730328213957375716542649984, 2.13064428244546652993171754803, 3.76434797227200077707919640544, 4.58838657056320703633670128927, 6.119356949456832656512220751710, 7.112628980026707230692422115450, 8.14573538973047058714189378339, 9.36746434828114778488372064381, 10.4700350196212049481224214758, 11.21029847652039630258621583399, 12.63335553227036713024693856642, 13.35455142925447921463580352559, 14.46336163613413856142716413012, 15.40553763219103385067247694614, 16.50330692783588430690027345388, 17.39639005070992714358944701634, 18.222830080685111104321817468, 19.47045981392231103797943017698, 20.26578768046850379216804104585, 21.04186471354191261241415142898, 22.231079335132232689967262998881, 23.206392069070179786789647584295, 23.771530394091276565977493151022, 24.94666945802999037445929107953, 26.04965978006639919169387527608