L(s) = 1 | + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.743 + 0.669i)5-s + (0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.5 + 0.866i)10-s + (0.913 + 0.406i)14-s + (0.669 − 0.743i)16-s + (0.669 − 0.743i)17-s + (0.406 − 0.913i)19-s + (−0.951 − 0.309i)20-s − 23-s + (0.104 + 0.994i)25-s + (−0.207 + 0.978i)28-s + (0.104 − 0.994i)29-s + ⋯ |
L(s) = 1 | + (0.207 + 0.978i)2-s + (−0.913 + 0.406i)4-s + (0.743 + 0.669i)5-s + (0.587 − 0.809i)7-s + (−0.587 − 0.809i)8-s + (−0.5 + 0.866i)10-s + (0.913 + 0.406i)14-s + (0.669 − 0.743i)16-s + (0.669 − 0.743i)17-s + (0.406 − 0.913i)19-s + (−0.951 − 0.309i)20-s − 23-s + (0.104 + 0.994i)25-s + (−0.207 + 0.978i)28-s + (0.104 − 0.994i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.759113796 + 0.4417998257i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.759113796 + 0.4417998257i\) |
\(L(1)\) |
\(\approx\) |
\(1.192896889 + 0.4714656735i\) |
\(L(1)\) |
\(\approx\) |
\(1.192896889 + 0.4714656735i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.207 + 0.978i)T \) |
| 5 | \( 1 + (0.743 + 0.669i)T \) |
| 7 | \( 1 + (0.587 - 0.809i)T \) |
| 17 | \( 1 + (0.669 - 0.743i)T \) |
| 19 | \( 1 + (0.406 - 0.913i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (0.104 - 0.994i)T \) |
| 31 | \( 1 + (-0.207 - 0.978i)T \) |
| 37 | \( 1 + (-0.406 - 0.913i)T \) |
| 41 | \( 1 + (0.587 + 0.809i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.994 + 0.104i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (-0.406 - 0.913i)T \) |
| 61 | \( 1 + (-0.309 - 0.951i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (0.743 + 0.669i)T \) |
| 73 | \( 1 + (0.587 - 0.809i)T \) |
| 79 | \( 1 + (-0.669 - 0.743i)T \) |
| 83 | \( 1 + (-0.207 + 0.978i)T \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 97 | \( 1 + (0.951 + 0.309i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.15101984939736753160542318568, −20.29195808159906492810669681180, −19.605976329915092296387254333237, −18.57776505018159255756341943212, −18.11180767893074625267744288574, −17.36578882461416325982920315208, −16.505828571732438129814718824222, −15.484059707812797063869162642045, −14.31917585804285055428845318118, −14.15724806903788366101396901664, −13.0069933699423107796161120319, −12.25501597165543683670227136737, −11.970936140320103660130702270327, −10.70185551055015060799296054561, −10.13929908392403534686752516339, −9.2195473939214720830628307867, −8.6072681369905652385225861226, −7.83637453146294609255022256844, −6.137151003784106814334850326813, −5.548536882277392231046201540865, −4.86507498714834338706772606663, −3.8572943707549825222517287806, −2.811124781292579030013573235520, −1.77805181023896282406080924088, −1.29298464813829842352967624006,
0.72079613665578499475505008227, 2.17248427839427717964797437508, 3.30293483364110055559671373001, 4.24282377266724344084439695379, 5.14348438085075587325733266494, 5.93163323418520013643809715776, 6.781736210889006616491967741562, 7.51714919082684002306355144108, 8.11665142529057885043388600662, 9.45053402058551769730759156932, 9.81075140882035268765447729243, 10.91807653695502849663779966761, 11.730630488406763488106752091473, 12.94000666039208990009891595760, 13.66371355332827270068321740164, 14.19050684468569747600112059398, 14.74933902048925900998646658877, 15.73375522812320098024894587025, 16.47592884881595242719924496793, 17.36796728932221152623723466284, 17.78414585870600193597334016983, 18.46068331943587247422436653654, 19.38013227973597680546719254049, 20.51881415530520154155427055338, 21.23039903346829835255873430243