L(s) = 1 | + (0.997 − 0.0697i)2-s + (0.990 − 0.139i)4-s + (−0.173 + 0.984i)5-s + (0.848 + 0.529i)7-s + (0.978 − 0.207i)8-s + (−0.104 + 0.994i)10-s + (−0.848 − 0.529i)11-s + (0.559 − 0.829i)13-s + (0.882 + 0.469i)14-s + (0.961 − 0.275i)16-s + (−0.309 + 0.951i)17-s + (0.913 − 0.406i)19-s + (−0.0348 + 0.999i)20-s + (−0.882 − 0.469i)22-s + (0.882 + 0.469i)23-s + ⋯ |
L(s) = 1 | + (0.997 − 0.0697i)2-s + (0.990 − 0.139i)4-s + (−0.173 + 0.984i)5-s + (0.848 + 0.529i)7-s + (0.978 − 0.207i)8-s + (−0.104 + 0.994i)10-s + (−0.848 − 0.529i)11-s + (0.559 − 0.829i)13-s + (0.882 + 0.469i)14-s + (0.961 − 0.275i)16-s + (−0.309 + 0.951i)17-s + (0.913 − 0.406i)19-s + (−0.0348 + 0.999i)20-s + (−0.882 − 0.469i)22-s + (0.882 + 0.469i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.782 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.782 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(4.847196079 + 1.694586461i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.847196079 + 1.694586461i\) |
\(L(1)\) |
\(\approx\) |
\(2.273995879 + 0.3555168342i\) |
\(L(1)\) |
\(\approx\) |
\(2.273995879 + 0.3555168342i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (0.997 - 0.0697i)T \) |
| 5 | \( 1 + (-0.173 + 0.984i)T \) |
| 7 | \( 1 + (0.848 + 0.529i)T \) |
| 11 | \( 1 + (-0.848 - 0.529i)T \) |
| 13 | \( 1 + (0.559 - 0.829i)T \) |
| 17 | \( 1 + (-0.309 + 0.951i)T \) |
| 19 | \( 1 + (0.913 - 0.406i)T \) |
| 23 | \( 1 + (0.882 + 0.469i)T \) |
| 29 | \( 1 + (0.997 - 0.0697i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.961 - 0.275i)T \) |
| 43 | \( 1 + (0.438 - 0.898i)T \) |
| 47 | \( 1 + (0.719 + 0.694i)T \) |
| 53 | \( 1 + (-0.669 - 0.743i)T \) |
| 59 | \( 1 + (0.997 + 0.0697i)T \) |
| 61 | \( 1 + (0.173 + 0.984i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.978 - 0.207i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (-0.615 + 0.788i)T \) |
| 83 | \( 1 + (-0.438 + 0.898i)T \) |
| 89 | \( 1 + (-0.669 + 0.743i)T \) |
| 97 | \( 1 + (0.0348 - 0.999i)T \) |
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
−21.695279147805787288886459067640, −20.97554212321769195054559073904, −20.4849582986782008901123098857, −19.97496104172528122802135418883, −18.68305862170846343751693453640, −17.731000088658161706844119437151, −16.75493469848217966379413884285, −16.146901623308526397689493834265, −15.44734158166318406781302364797, −14.41994482015880518856435573715, −13.67963886962069237069366854458, −13.08042518342283428157488604382, −12.08765907647073903507212877850, −11.48711015639102514237494721950, −10.63329599369678794966057082207, −9.49885656073894562781619893727, −8.32910596539102360294188305792, −7.60212894673363698600861027093, −6.75755274331591844031422473645, −5.48770828481491929483785090639, −4.74534528940574617736218652472, −4.30678924402014286998286237547, −3.01121596904315556152456854948, −1.82573323185217122039690547085, −0.8944608140414140450074726605,
1.11985060508430339940994364114, 2.45627988389702232854732781239, 3.04428435676739507015209526768, 4.04458473625802196863730204372, 5.26277832124516671885213900571, 5.78076119631161289282211354429, 6.83269290204174692494916940864, 7.7602779603505772866580408101, 8.46542630680310439758018556296, 10.06938411770252756421057553516, 10.93723528434476911990134916826, 11.25712972220395307217146608875, 12.27476255372231571727045851409, 13.2560706843293794151390339461, 13.90540503338863272535215761347, 14.81748316577996210721165941786, 15.43104927881289167889378003705, 15.901708757269407425665547005596, 17.31397688814587676275551822768, 18.104403751546629792722234125805, 18.89468402636976816718745245185, 19.73418863591266523056891856405, 20.72266252267823032251318571340, 21.36180828310515038421023312079, 22.04914731834406289870928813571