L(s) = 1 | + (0.241 − 0.970i)2-s + (−0.0348 + 0.999i)3-s + (−0.882 − 0.469i)4-s + (0.438 − 0.898i)5-s + (0.961 + 0.275i)6-s + (0.669 + 0.743i)7-s + (−0.669 + 0.743i)8-s + (−0.997 − 0.0697i)9-s + (−0.766 − 0.642i)10-s + (0.5 − 0.866i)12-s + (−0.559 + 0.829i)13-s + (0.882 − 0.469i)14-s + (0.882 + 0.469i)15-s + (0.559 + 0.829i)16-s + (−0.997 + 0.0697i)17-s + (−0.309 + 0.951i)18-s + ⋯ |
L(s) = 1 | + (0.241 − 0.970i)2-s + (−0.0348 + 0.999i)3-s + (−0.882 − 0.469i)4-s + (0.438 − 0.898i)5-s + (0.961 + 0.275i)6-s + (0.669 + 0.743i)7-s + (−0.669 + 0.743i)8-s + (−0.997 − 0.0697i)9-s + (−0.766 − 0.642i)10-s + (0.5 − 0.866i)12-s + (−0.559 + 0.829i)13-s + (0.882 − 0.469i)14-s + (0.882 + 0.469i)15-s + (0.559 + 0.829i)16-s + (−0.997 + 0.0697i)17-s + (−0.309 + 0.951i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00483 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00483 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6617799918 + 0.6649869282i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6617799918 + 0.6649869282i\) |
\(L(1)\) |
\(\approx\) |
\(0.9559589656 - 0.08658261782i\) |
\(L(1)\) |
\(\approx\) |
\(0.9559589656 - 0.08658261782i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.241 - 0.970i)T \) |
| 3 | \( 1 + (-0.0348 + 0.999i)T \) |
| 5 | \( 1 + (0.438 - 0.898i)T \) |
| 7 | \( 1 + (0.669 + 0.743i)T \) |
| 13 | \( 1 + (-0.559 + 0.829i)T \) |
| 17 | \( 1 + (-0.997 + 0.0697i)T \) |
| 23 | \( 1 + (0.173 - 0.984i)T \) |
| 29 | \( 1 + (-0.848 + 0.529i)T \) |
| 31 | \( 1 + (0.104 + 0.994i)T \) |
| 37 | \( 1 + (-0.309 + 0.951i)T \) |
| 41 | \( 1 + (-0.0348 + 0.999i)T \) |
| 43 | \( 1 + (0.173 + 0.984i)T \) |
| 47 | \( 1 + (-0.374 + 0.927i)T \) |
| 53 | \( 1 + (-0.438 - 0.898i)T \) |
| 59 | \( 1 + (0.374 + 0.927i)T \) |
| 61 | \( 1 + (-0.719 + 0.694i)T \) |
| 67 | \( 1 + (-0.766 - 0.642i)T \) |
| 71 | \( 1 + (-0.438 + 0.898i)T \) |
| 73 | \( 1 + (-0.615 + 0.788i)T \) |
| 79 | \( 1 + (-0.961 + 0.275i)T \) |
| 83 | \( 1 + (0.913 - 0.406i)T \) |
| 89 | \( 1 + (0.939 - 0.342i)T \) |
| 97 | \( 1 + (0.241 - 0.970i)T \) |
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
−26.140194482065902247733807046042, −25.133425532915733097461941689440, −24.44633341022816565738715187188, −23.57105317787541646194635298086, −22.71638086747131280407189420329, −22.00204575016428392608219912284, −20.60148662609698691858977554239, −19.33407916979542935360989276362, −18.308940852778249296460602353452, −17.49080619981214209564273232148, −17.14267117588276447209284520064, −15.4034350042986880716097193417, −14.5823108794614714355694741744, −13.686300647184193693828654033136, −13.11647544010475147556254779022, −11.70189901546911097776498084922, −10.55374682325847952399584627720, −9.11724631877101722095343684851, −7.684314943795078460898682295, −7.320973533134985118445785899664, −6.199647210981747115775119557165, −5.2171534613247796274992581419, −3.61385725118613346575332699610, −2.131016158017934588993550101261, −0.281327478828413856308576440981,
1.633133429957148294317810813896, 2.79960966057081605471824607434, 4.507385635708892942437026896713, 4.83023444948920105448658936014, 6.03780825319588834125658026628, 8.514631112591705042080141394530, 9.0342132847957648975133667162, 9.96501931790190420633195985973, 11.13310347508765026763355515978, 11.92389194616997579413924443469, 12.9627244946296029673862201159, 14.203903297998023526452859591580, 14.930738307068357945364845127850, 16.21223761995563842187589679676, 17.2698392594519395626918588000, 18.14131926155761401403970606029, 19.48950553615599906753801937815, 20.400094698974209764668579364318, 21.13623070543060076295189576991, 21.749571845791753070997578520, 22.511012662640347851145176228601, 23.89801001155936120908253817718, 24.646402645462418535037702516675, 26.06068368468386312653386268301, 27.08361591004376381043358619900