L(s) = 1 | + (0.793 − 0.608i)3-s + (−0.608 + 0.793i)5-s + (0.258 − 0.965i)9-s + (−0.130 + 0.991i)11-s + (−0.382 + 0.923i)13-s + i·15-s + (−0.866 − 0.5i)17-s + (0.991 − 0.130i)19-s + (0.258 − 0.965i)23-s + (−0.258 − 0.965i)25-s + (−0.382 − 0.923i)27-s + (−0.923 − 0.382i)29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (0.608 − 0.793i)37-s + ⋯ |
L(s) = 1 | + (0.793 − 0.608i)3-s + (−0.608 + 0.793i)5-s + (0.258 − 0.965i)9-s + (−0.130 + 0.991i)11-s + (−0.382 + 0.923i)13-s + i·15-s + (−0.866 − 0.5i)17-s + (0.991 − 0.130i)19-s + (0.258 − 0.965i)23-s + (−0.258 − 0.965i)25-s + (−0.382 − 0.923i)27-s + (−0.923 − 0.382i)29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (0.608 − 0.793i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03475634067 - 0.3049074328i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03475634067 - 0.3049074328i\) |
\(L(1)\) |
\(\approx\) |
\(0.9731554966 - 0.07809752118i\) |
\(L(1)\) |
\(\approx\) |
\(0.9731554966 - 0.07809752118i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.793 - 0.608i)T \) |
| 5 | \( 1 + (-0.608 + 0.793i)T \) |
| 11 | \( 1 + (-0.130 + 0.991i)T \) |
| 13 | \( 1 + (-0.382 + 0.923i)T \) |
| 17 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (0.991 - 0.130i)T \) |
| 23 | \( 1 + (0.258 - 0.965i)T \) |
| 29 | \( 1 + (-0.923 - 0.382i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.608 - 0.793i)T \) |
| 41 | \( 1 + (-0.707 - 0.707i)T \) |
| 43 | \( 1 + (-0.923 + 0.382i)T \) |
| 47 | \( 1 + (-0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.130 - 0.991i)T \) |
| 59 | \( 1 + (-0.991 - 0.130i)T \) |
| 61 | \( 1 + (-0.130 - 0.991i)T \) |
| 67 | \( 1 + (0.793 - 0.608i)T \) |
| 71 | \( 1 + (-0.707 + 0.707i)T \) |
| 73 | \( 1 + (-0.965 + 0.258i)T \) |
| 79 | \( 1 + (0.866 - 0.5i)T \) |
| 83 | \( 1 + (-0.382 + 0.923i)T \) |
| 89 | \( 1 + (0.965 + 0.258i)T \) |
| 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
−24.40731636996071808378778271975, −23.46406117086787221272598176141, −22.21066889282572202527124634164, −21.66751131054358498495864760782, −20.49711195510083193521752121608, −20.08998661338168007357715188767, −19.317476609489461842867363692226, −18.327924817967606855475978336870, −16.97707748397390343707243540031, −16.32871061145071917905559143734, −15.40069660189790848720403903108, −14.90856300623057187897969059708, −13.462714948688898304482600135016, −13.15286392988977032506081432724, −11.748631295843149413834074493560, −10.9183858927078903982828281329, −9.78828371299668082857738950862, −8.93534921864135370177936035057, −8.13644179256204666367337437678, −7.44017543560949372608306286915, −5.68902446844682845867176973436, −4.86378427000631663242665587404, −3.73695216430093637769041852777, −3.00515395667868989678696928753, −1.45223879554419200844098703297,
0.068785606662983916984052174489, 1.82201144534934561384763956387, 2.687010300975251957015495310741, 3.76632851756202553310860055045, 4.81545177582244077618376680899, 6.60932300644672728625653788097, 7.08827552665980217411443265193, 7.89951477218240233284305386218, 9.06794023637020465517471588369, 9.850663674264121766649232910700, 11.14915486678470338915519767660, 11.9811629015561722223310855399, 12.85895587609333945932398827375, 13.93473284037072047008103145378, 14.6289904725005580465271279397, 15.330435424593660543903890398982, 16.30214521317891222875679236390, 17.72292787726764559201157526264, 18.33814526128454608196153188150, 19.09119881644300648959124403488, 19.97764949885596911660583431713, 20.50412724186597576030103622067, 21.749708188271492279419623198103, 22.67348160101241182634107367943, 23.42389674423380607821369315794