L(s) = 1 | + (0.984 − 0.173i)3-s + (0.642 + 0.766i)5-s + (0.5 − 0.866i)7-s + (0.939 − 0.342i)9-s + (0.866 − 0.5i)11-s + (−0.984 − 0.173i)13-s + (0.766 + 0.642i)15-s + (−0.939 − 0.342i)17-s + (0.342 − 0.939i)21-s + (−0.766 − 0.642i)23-s + (−0.173 + 0.984i)25-s + (0.866 − 0.5i)27-s + (0.342 + 0.939i)29-s + (−0.5 + 0.866i)31-s + (0.766 − 0.642i)33-s + ⋯ |
L(s) = 1 | + (0.984 − 0.173i)3-s + (0.642 + 0.766i)5-s + (0.5 − 0.866i)7-s + (0.939 − 0.342i)9-s + (0.866 − 0.5i)11-s + (−0.984 − 0.173i)13-s + (0.766 + 0.642i)15-s + (−0.939 − 0.342i)17-s + (0.342 − 0.939i)21-s + (−0.766 − 0.642i)23-s + (−0.173 + 0.984i)25-s + (0.866 − 0.5i)27-s + (0.342 + 0.939i)29-s + (−0.5 + 0.866i)31-s + (0.766 − 0.642i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.002236422 - 0.2353078418i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.002236422 - 0.2353078418i\) |
\(L(1)\) |
\(\approx\) |
\(1.613989329 - 0.1015644536i\) |
\(L(1)\) |
\(\approx\) |
\(1.613989329 - 0.1015644536i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.984 - 0.173i)T \) |
| 5 | \( 1 + (0.642 + 0.766i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (-0.984 - 0.173i)T \) |
| 17 | \( 1 + (-0.939 - 0.342i)T \) |
| 23 | \( 1 + (-0.766 - 0.642i)T \) |
| 29 | \( 1 + (0.342 + 0.939i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.642 + 0.766i)T \) |
| 47 | \( 1 + (-0.939 + 0.342i)T \) |
| 53 | \( 1 + (-0.642 + 0.766i)T \) |
| 59 | \( 1 + (0.342 - 0.939i)T \) |
| 61 | \( 1 + (0.642 - 0.766i)T \) |
| 67 | \( 1 + (0.342 + 0.939i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (-0.173 - 0.984i)T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + (0.866 + 0.5i)T \) |
| 89 | \( 1 + (-0.173 + 0.984i)T \) |
| 97 | \( 1 + (-0.939 - 0.342i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−25.214490914575637160119866189501, −24.60333683179691749469808377260, −24.07044261295297038545501818918, −22.26621008660710070053013716809, −21.68210838544061959298181836105, −20.881519174927052300827639212580, −19.94365480610580405459001095994, −19.33796314569477473619735437546, −18.01081508239026567204877320426, −17.31073032753019569564642656034, −16.13268730440212140996791689062, −15.09873541996689572290366451480, −14.45457478900011470746351397054, −13.44241308008846474498815399483, −12.526471624231699516132703472, −11.607866692613511185971977997866, −9.965511572578000957874385048721, −9.31281578227166017213954045746, −8.582915252406065637453506481799, −7.53634305673234688455553456846, −6.15543518001286975947369675013, −4.898944190910744111030160079412, −4.06467783805031469059946567817, −2.33256815197859557269887051371, −1.78351923760303978753739550387,
1.46976944706219825581550934304, 2.57850429775204221431844316671, 3.658126589837528278547794358564, 4.817304679767231657609578201957, 6.535681154771819113276271965874, 7.140495997874810961147314675504, 8.270392492871639244415740797279, 9.34544966681603189719221130213, 10.23014999618342163119084448003, 11.1643674936033230621045502818, 12.521389836865434518004267018621, 13.67733518185630353378022252133, 14.26096806863964469364755927807, 14.79214182114831295909043730131, 16.12464350884212480546935018701, 17.35701203494621305896175428115, 18.01414023253354727146339550589, 19.112943352030215023009945020655, 19.91364512599160682525099766515, 20.63378466695013931361959378136, 21.80784717654456828782375442224, 22.307765221888415147911959568977, 23.75275997481746999215873077108, 24.54261301725962667843020620134, 25.24896131885218411094678184990