L(s) = 1 | + (0.438 − 0.898i)2-s + (−0.615 − 0.788i)4-s + (0.997 + 0.0697i)5-s + (−0.0348 + 0.999i)7-s + (−0.978 + 0.207i)8-s + (0.5 − 0.866i)10-s + (0.719 − 0.694i)13-s + (0.882 + 0.469i)14-s + (−0.241 + 0.970i)16-s + (−0.104 − 0.994i)17-s + (0.978 − 0.207i)19-s + (−0.559 − 0.829i)20-s + (0.939 − 0.342i)23-s + (0.990 + 0.139i)25-s + (−0.309 − 0.951i)26-s + ⋯ |
L(s) = 1 | + (0.438 − 0.898i)2-s + (−0.615 − 0.788i)4-s + (0.997 + 0.0697i)5-s + (−0.0348 + 0.999i)7-s + (−0.978 + 0.207i)8-s + (0.5 − 0.866i)10-s + (0.719 − 0.694i)13-s + (0.882 + 0.469i)14-s + (−0.241 + 0.970i)16-s + (−0.104 − 0.994i)17-s + (0.978 − 0.207i)19-s + (−0.559 − 0.829i)20-s + (0.939 − 0.342i)23-s + (0.990 + 0.139i)25-s + (−0.309 − 0.951i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.404527172 - 1.031596089i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.404527172 - 1.031596089i\) |
\(L(1)\) |
\(\approx\) |
\(1.292779712 - 0.6443199199i\) |
\(L(1)\) |
\(\approx\) |
\(1.292779712 - 0.6443199199i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.438 - 0.898i)T \) |
| 5 | \( 1 + (0.997 + 0.0697i)T \) |
| 7 | \( 1 + (-0.0348 + 0.999i)T \) |
| 13 | \( 1 + (0.719 - 0.694i)T \) |
| 17 | \( 1 + (-0.104 - 0.994i)T \) |
| 19 | \( 1 + (0.978 - 0.207i)T \) |
| 23 | \( 1 + (0.939 - 0.342i)T \) |
| 29 | \( 1 + (-0.882 + 0.469i)T \) |
| 31 | \( 1 + (0.961 + 0.275i)T \) |
| 37 | \( 1 + (-0.978 - 0.207i)T \) |
| 41 | \( 1 + (-0.882 - 0.469i)T \) |
| 43 | \( 1 + (-0.766 + 0.642i)T \) |
| 47 | \( 1 + (0.615 - 0.788i)T \) |
| 53 | \( 1 + (0.809 + 0.587i)T \) |
| 59 | \( 1 + (0.374 - 0.927i)T \) |
| 61 | \( 1 + (-0.961 + 0.275i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + (0.104 + 0.994i)T \) |
| 73 | \( 1 + (-0.669 + 0.743i)T \) |
| 79 | \( 1 + (-0.438 + 0.898i)T \) |
| 83 | \( 1 + (-0.719 - 0.694i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.997 + 0.0697i)T \) |
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
−25.638232048285094470202873117617, −24.6097100891218613750215356775, −23.89079109649722090696822277543, −22.995046188338837900714076840819, −22.1584220960854496645884268504, −21.15367443421065987502799495674, −20.58088489823186163419902704749, −19.05351547556389238656507420759, −18.01892928714355420993773417313, −17.08934513654327840314593731497, −16.68022316255002883048719711391, −15.48778689941938777365271501858, −14.42937215311432251862820112088, −13.53715449380780178434025547108, −13.221025371408769446500018799421, −11.79084613461161507743047806408, −10.46723916029688932507132805876, −9.44390856308633985164940805859, −8.45662798265228823780723932119, −7.239679410050579366192473851955, −6.411197119588251285109180317724, −5.46285395288498208666982786550, −4.31463193527452718863632950722, −3.26094274599546926700751653736, −1.45614762242950874673891094400,
1.26134056654454969141545415057, 2.51526720876764470313530530127, 3.28593872489525045527563746834, 5.0796732943432136998538634251, 5.54403129187255697737579686477, 6.74552266195831832969460972301, 8.62966329182713257554747216320, 9.33919071620117747342542858493, 10.27947179420189165000449253591, 11.28466514811109544324604993023, 12.23662699943480037887264845209, 13.21352662444693770301163704599, 13.870188022139380651270202112441, 14.9422903744145244951133498777, 15.842702032987540869749524176657, 17.34551058761012307257895078890, 18.360463317018568972814650361582, 18.63563246160129511939117219823, 20.10182150966512939617901181079, 20.8120017860996335130707046615, 21.588487217351474255464005801338, 22.41667889412787230075649680363, 23.00989551277378193834082729010, 24.519414820613054147537727379514, 24.95471072347500763950349175852