L(s) = 1 | + (−0.798 − 0.601i)2-s + (−0.683 − 0.729i)3-s + (0.276 + 0.961i)4-s + (−0.150 + 0.988i)5-s + (0.107 + 0.994i)6-s + (−0.744 + 0.668i)7-s + (0.357 − 0.933i)8-s + (−0.0645 + 0.997i)9-s + (0.714 − 0.699i)10-s + (−0.397 + 0.917i)11-s + (0.512 − 0.858i)12-s + (−0.976 + 0.213i)13-s + (0.996 − 0.0859i)14-s + (0.823 − 0.566i)15-s + (−0.847 + 0.530i)16-s + (−0.798 − 0.601i)17-s + ⋯ |
L(s) = 1 | + (−0.798 − 0.601i)2-s + (−0.683 − 0.729i)3-s + (0.276 + 0.961i)4-s + (−0.150 + 0.988i)5-s + (0.107 + 0.994i)6-s + (−0.744 + 0.668i)7-s + (0.357 − 0.933i)8-s + (−0.0645 + 0.997i)9-s + (0.714 − 0.699i)10-s + (−0.397 + 0.917i)11-s + (0.512 − 0.858i)12-s + (−0.976 + 0.213i)13-s + (0.996 − 0.0859i)14-s + (0.823 − 0.566i)15-s + (−0.847 + 0.530i)16-s + (−0.798 − 0.601i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03601104790 - 0.1507011351i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03601104790 - 0.1507011351i\) |
\(L(1)\) |
\(\approx\) |
\(0.3969029835 - 0.09661642126i\) |
\(L(1)\) |
\(\approx\) |
\(0.3969029835 - 0.09661642126i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 293 | \( 1 \) |
good | 2 | \( 1 + (-0.798 - 0.601i)T \) |
| 3 | \( 1 + (-0.683 - 0.729i)T \) |
| 5 | \( 1 + (-0.150 + 0.988i)T \) |
| 7 | \( 1 + (-0.744 + 0.668i)T \) |
| 11 | \( 1 + (-0.397 + 0.917i)T \) |
| 13 | \( 1 + (-0.976 + 0.213i)T \) |
| 17 | \( 1 + (-0.798 - 0.601i)T \) |
| 19 | \( 1 + (0.823 - 0.566i)T \) |
| 23 | \( 1 + (0.107 - 0.994i)T \) |
| 29 | \( 1 + (0.0215 - 0.999i)T \) |
| 31 | \( 1 + (-0.150 - 0.988i)T \) |
| 37 | \( 1 + (0.436 - 0.899i)T \) |
| 41 | \( 1 + (0.584 - 0.811i)T \) |
| 43 | \( 1 + (-0.890 + 0.455i)T \) |
| 47 | \( 1 + (0.869 + 0.493i)T \) |
| 53 | \( 1 + (-0.397 + 0.917i)T \) |
| 59 | \( 1 + (-0.474 + 0.880i)T \) |
| 61 | \( 1 + (-0.925 - 0.377i)T \) |
| 67 | \( 1 + (-0.548 + 0.835i)T \) |
| 71 | \( 1 + (0.941 + 0.337i)T \) |
| 73 | \( 1 + (0.276 - 0.961i)T \) |
| 79 | \( 1 + (-0.925 - 0.377i)T \) |
| 83 | \( 1 + (0.772 - 0.635i)T \) |
| 89 | \( 1 + (-0.925 + 0.377i)T \) |
| 97 | \( 1 + (0.985 - 0.171i)T \) |
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
−26.04758634371755185125216613910, −24.949020157024509230505955710662, −23.92573266197059108692165455965, −23.4998599251690845288306501519, −22.330408723872800486299979415964, −21.32846591742902134156515650251, −20.06477093953590987111993440867, −19.725793187006083107255515686, −18.332041294687230828323786147529, −17.250853029066366282440609389104, −16.71413332589381950757976838093, −16.029679736636741101075772879755, −15.33624461197506788197767326073, −13.9864163726237931929939753421, −12.80678793520058580552069457681, −11.60528152794667729857530186059, −10.57682990591672274923404295675, −9.76962637510826417420792560765, −8.9718518579348554807521371812, −7.84471504629138582188611283982, −6.652635720323283711292057133629, −5.5754677719135608065157776552, −4.81376796800239455441035961703, −3.40586193919832512750640577035, −1.16794220520865109818238027127,
0.15950470755051451943742691249, 2.29938659689415290250579467226, 2.65330871239211408018916407873, 4.50818461931110105766813700250, 6.12791413289488137156141477425, 7.124411833261797411256293291605, 7.617741832479012362094879795637, 9.22971961296209826681803535473, 10.07812036491916789521132416571, 11.08470508997690034727210275348, 11.92386748495212737394862594484, 12.61556078865824825517288645681, 13.64358907125705709573680384797, 15.19894171073380021957597086069, 16.07891687163094612515896121675, 17.2280808819908944486634949815, 18.04555742919685334332932077574, 18.608777565646446200425324112726, 19.38679277035024789577578012682, 20.20263574806432380021742338674, 21.69677141950258328627531345763, 22.43028358144513222324571477318, 22.84831554769053697766336065128, 24.41712048953136180338969326686, 25.14233225488919339834408401404