L(s) = 1 | + (−0.877 − 0.480i)2-s + (0.113 − 0.993i)3-s + (0.538 + 0.842i)4-s + (0.613 − 0.789i)5-s + (−0.576 + 0.816i)6-s + (0.995 − 0.0909i)7-s + (−0.0682 − 0.997i)8-s + (−0.974 − 0.225i)9-s + (−0.917 + 0.398i)10-s + (0.934 + 0.356i)11-s + (0.898 − 0.439i)12-s + (0.803 + 0.595i)13-s + (−0.917 − 0.398i)14-s + (−0.715 − 0.699i)15-s + (−0.419 + 0.907i)16-s + (0.983 + 0.181i)17-s + ⋯ |
L(s) = 1 | + (−0.877 − 0.480i)2-s + (0.113 − 0.993i)3-s + (0.538 + 0.842i)4-s + (0.613 − 0.789i)5-s + (−0.576 + 0.816i)6-s + (0.995 − 0.0909i)7-s + (−0.0682 − 0.997i)8-s + (−0.974 − 0.225i)9-s + (−0.917 + 0.398i)10-s + (0.934 + 0.356i)11-s + (0.898 − 0.439i)12-s + (0.803 + 0.595i)13-s + (−0.917 − 0.398i)14-s + (−0.715 − 0.699i)15-s + (−0.419 + 0.907i)16-s + (0.983 + 0.181i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6300366774 - 0.6993078987i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6300366774 - 0.6993078987i\) |
\(L(1)\) |
\(\approx\) |
\(0.7637694890 - 0.4946288936i\) |
\(L(1)\) |
\(\approx\) |
\(0.7637694890 - 0.4946288936i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 139 | \( 1 \) |
good | 2 | \( 1 + (-0.877 - 0.480i)T \) |
| 3 | \( 1 + (0.113 - 0.993i)T \) |
| 5 | \( 1 + (0.613 - 0.789i)T \) |
| 7 | \( 1 + (0.995 - 0.0909i)T \) |
| 11 | \( 1 + (0.934 + 0.356i)T \) |
| 13 | \( 1 + (0.803 + 0.595i)T \) |
| 17 | \( 1 + (0.983 + 0.181i)T \) |
| 19 | \( 1 + (-0.648 + 0.761i)T \) |
| 23 | \( 1 + (-0.576 - 0.816i)T \) |
| 29 | \( 1 + (-0.998 + 0.0455i)T \) |
| 31 | \( 1 + (-0.974 + 0.225i)T \) |
| 37 | \( 1 + (-0.715 + 0.699i)T \) |
| 41 | \( 1 + (-0.949 - 0.313i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.377 - 0.926i)T \) |
| 53 | \( 1 + (0.0227 + 0.999i)T \) |
| 59 | \( 1 + (0.460 + 0.887i)T \) |
| 61 | \( 1 + (-0.949 + 0.313i)T \) |
| 67 | \( 1 + (-0.158 + 0.987i)T \) |
| 71 | \( 1 + (0.377 + 0.926i)T \) |
| 73 | \( 1 + (-0.829 + 0.557i)T \) |
| 79 | \( 1 + (0.203 - 0.979i)T \) |
| 83 | \( 1 + (-0.158 - 0.987i)T \) |
| 89 | \( 1 + (-0.247 + 0.968i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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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
−28.091118551547266390570630838221, −27.654833463912051711026607190218, −26.78362394083396898366622024085, −25.74064657883105807515067387655, −25.24139407916324628222839626276, −23.89042905532081763275053751800, −22.67913294746485493593470704894, −21.557172097625323672737858949043, −20.722280953116012112182729596786, −19.59565672300931148403701158776, −18.37722939322873716507673915809, −17.49435082947609205295397276280, −16.67949472633042270638405353460, −15.39414978321204384912707452770, −14.65261602621927317131428491997, −13.8988225546395480875135470754, −11.39434685560463612541628831739, −10.87511128770597042800412501347, −9.74820199114927537189107850745, −8.8351569720366528126550027481, −7.69422627050140248209770876675, −6.16214777276610483370272480935, −5.27221248110168882517862466777, −3.44911216370210937659139722554, −1.7908198802445061052479022593,
1.39437634324382771595631473585, 1.8859825407557914143535306494, 3.92380014235499336413789144961, 5.8304584287769823497751195710, 7.09758854276105651901002903798, 8.36166899183045692720004450193, 8.907897008112983722345197226940, 10.37178844580392554507098859183, 11.755727706922988782126596849808, 12.35165762758737360603023196656, 13.5865134183313445814256735126, 14.651895092787986995618305394555, 16.68990684703635477727721670442, 17.08762568597554962584678302909, 18.212309577313402207885336188566, 18.88620736238965121121379322239, 20.27556288036294509886448927225, 20.68608753398434242482870714488, 21.872096026737034170611414178637, 23.5167355015166372379908294290, 24.48795361725824360154740666087, 25.25251541701432763510665250341, 25.98665462411568490274938321661, 27.55132581442466061407856252516, 28.12859176977807723428427658754