L(s) = 1 | + (−0.104 − 0.994i)3-s + (−0.978 + 0.207i)9-s + (−0.978 − 0.207i)11-s + (−0.309 + 0.951i)13-s + (0.913 − 0.406i)17-s + (0.104 − 0.994i)19-s + (0.669 − 0.743i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (0.913 − 0.406i)31-s + (−0.104 + 0.994i)33-s + (−0.978 + 0.207i)37-s + (0.978 + 0.207i)39-s + (−0.309 + 0.951i)41-s − 43-s + ⋯ |
L(s) = 1 | + (−0.104 − 0.994i)3-s + (−0.978 + 0.207i)9-s + (−0.978 − 0.207i)11-s + (−0.309 + 0.951i)13-s + (0.913 − 0.406i)17-s + (0.104 − 0.994i)19-s + (0.669 − 0.743i)23-s + (0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (0.913 − 0.406i)31-s + (−0.104 + 0.994i)33-s + (−0.978 + 0.207i)37-s + (0.978 + 0.207i)39-s + (−0.309 + 0.951i)41-s − 43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.06032722573 - 0.7231296027i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06032722573 - 0.7231296027i\) |
\(L(1)\) |
\(\approx\) |
\(0.7551493780 - 0.3615302254i\) |
\(L(1)\) |
\(\approx\) |
\(0.7551493780 - 0.3615302254i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.104 - 0.994i)T \) |
| 11 | \( 1 + (-0.978 - 0.207i)T \) |
| 13 | \( 1 + (-0.309 + 0.951i)T \) |
| 17 | \( 1 + (0.913 - 0.406i)T \) |
| 19 | \( 1 + (0.104 - 0.994i)T \) |
| 23 | \( 1 + (0.669 - 0.743i)T \) |
| 29 | \( 1 + (0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.913 - 0.406i)T \) |
| 37 | \( 1 + (-0.978 + 0.207i)T \) |
| 41 | \( 1 + (-0.309 + 0.951i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.913 - 0.406i)T \) |
| 53 | \( 1 + (-0.104 - 0.994i)T \) |
| 59 | \( 1 + (-0.669 - 0.743i)T \) |
| 61 | \( 1 + (0.669 - 0.743i)T \) |
| 67 | \( 1 + (-0.913 + 0.406i)T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.978 - 0.207i)T \) |
| 79 | \( 1 + (-0.913 - 0.406i)T \) |
| 83 | \( 1 + (-0.809 - 0.587i)T \) |
| 89 | \( 1 + (-0.669 + 0.743i)T \) |
| 97 | \( 1 + (-0.809 + 0.587i)T \) |
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
−21.14903400080120244965561085869, −20.60040227279738862367676633267, −19.77345390030116504931930949613, −18.96094472701720904813768084227, −17.97952830577350528929672518282, −17.31560488062042742100061900655, −16.5693721430601140418556597184, −15.73582959869709883568760718784, −15.24893358295533762431273212610, −14.4552170348118356121345412688, −13.6512784008424968220025868068, −12.57354839727648729991074666499, −12.00208842143016298326021487415, −10.88096616158056852663954572190, −10.21782783680426936130480574456, −9.87021461684355735404679876878, −8.62829971721688440187374190473, −8.05002368900996479797389640505, −7.08147625354691051397987965011, −5.7290510513398300636169980746, −5.35894985147844819705657726660, −4.452748890212353494112629196592, −3.31969470107654114362598499742, −2.86151980951675746006743652098, −1.35112236630910954876666836323,
0.28697460734766304268748916483, 1.46141001335891750720498041055, 2.53602645917418698820222603178, 3.139384557629580002000389485782, 4.70039646234702340696077262750, 5.277127987200873346829794917710, 6.46796990374365798254967650805, 6.92516811055911783636932716818, 7.93499900914260897426706831701, 8.48177926724997858596982135879, 9.54716707094079460198588574332, 10.43348842075179333805835558814, 11.47348586914808354092783465450, 11.90154026960417207180510875450, 12.88439917228617204237983583774, 13.48777898141428218293061157246, 14.16690762260348447806093902742, 15.01837407108410538852342350963, 16.021156388925326686325785707803, 16.76419247347589795818824744109, 17.49591274674932646255406488474, 18.29506685872014606640180535854, 18.94274180695903502357548218028, 19.41399232419520577346991469903, 20.434350051979591772665887555837