L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.951 − 0.309i)3-s + (0.809 − 0.587i)4-s + (−0.809 + 0.587i)6-s + (0.951 + 0.309i)7-s + (−0.587 + 0.809i)8-s + (0.809 − 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.587 − 0.809i)12-s + (−0.951 − 0.309i)13-s − 14-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + (−0.587 + 0.809i)18-s + (−0.309 + 0.951i)19-s + ⋯ |
L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.951 − 0.309i)3-s + (0.809 − 0.587i)4-s + (−0.809 + 0.587i)6-s + (0.951 + 0.309i)7-s + (−0.587 + 0.809i)8-s + (0.809 − 0.587i)9-s + (−0.809 − 0.587i)11-s + (0.587 − 0.809i)12-s + (−0.951 − 0.309i)13-s − 14-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + (−0.587 + 0.809i)18-s + (−0.309 + 0.951i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1333273773 - 0.6542167094i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1333273773 - 0.6542167094i\) |
\(L(1)\) |
\(\approx\) |
\(0.8557404125 - 0.1133028792i\) |
\(L(1)\) |
\(\approx\) |
\(0.8557404125 - 0.1133028792i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 3 | \( 1 + (0.951 - 0.309i)T \) |
| 7 | \( 1 + (0.951 + 0.309i)T \) |
| 11 | \( 1 + (-0.809 - 0.587i)T \) |
| 13 | \( 1 + (-0.951 - 0.309i)T \) |
| 17 | \( 1 + (0.587 - 0.809i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 - T \) |
| 37 | \( 1 + (-0.951 + 0.309i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + (0.587 + 0.809i)T \) |
| 47 | \( 1 + (-0.951 + 0.309i)T \) |
| 53 | \( 1 + (-0.951 - 0.309i)T \) |
| 59 | \( 1 + (-0.309 + 0.951i)T \) |
| 61 | \( 1 + (0.309 - 0.951i)T \) |
| 67 | \( 1 + (-0.587 - 0.809i)T \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.587 + 0.809i)T \) |
| 79 | \( 1 + (-0.309 + 0.951i)T \) |
| 83 | \( 1 + (-0.951 - 0.309i)T \) |
| 89 | \( 1 + (0.809 + 0.587i)T \) |
| 97 | \( 1 + (-0.587 - 0.809i)T \) |
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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
−22.01611268361214169783181153487, −21.39280264803754980724553858948, −20.75184988792960028638634846623, −20.0826029804331036983582495368, −19.33149843289176357735583806645, −18.66051786726221712015216073589, −17.601798061351991683090935920780, −17.128860872010122012487485588906, −16.007245912490730326538099580858, −15.15079903936555082382314261443, −14.66039032561078528276239802697, −13.45468216006479842135530974224, −12.63354982111134586321959102274, −11.58855361397274169588448731849, −10.64463312527370522300818311748, −10.00937043715326663766921275834, −9.18826693386401584758439783887, −8.303256002635046238677673197664, −7.57752232716224172457563917622, −7.06346903507813453810261988401, −5.29882925893755434050327153639, −4.29582874691580868320998982664, −3.229125442545808376048033150011, −2.16169207810719887130794390140, −1.55349556686568168143517449549,
0.16079243248200308980422270336, 1.458819618705965389783494935478, 2.33941655455974471010046098547, 3.176951651987220123368313351097, 4.81222342367694294882337503888, 5.72130380030717963401216141756, 6.94680996747218336672181937733, 7.87447232134617045326257874121, 8.150013336099734021012956634722, 9.12431284445337712287408918492, 9.98963434434379254054288823907, 10.785860742391822077287915208990, 11.88494527295591082042755297804, 12.67455037950831399389015324154, 14.00749848864104312038915823565, 14.56179356074591357824417794032, 15.24249685906562151626110437456, 16.141112639943084057293908813779, 17.03794796842101414041640388635, 18.01530054146254643752005757695, 18.64518251898141201589813315991, 19.094851128733786523407423161536, 20.215342989912578283670350074240, 20.79054918132444639539672707209, 21.31838078545059417082381018258