L(s) = 1 | + (0.725 − 0.687i)2-s + (0.976 − 0.214i)3-s + (0.0541 − 0.998i)4-s + (−0.161 + 0.986i)5-s + (0.561 − 0.827i)6-s + (0.468 − 0.883i)7-s + (−0.647 − 0.762i)8-s + (0.907 − 0.419i)9-s + (0.561 + 0.827i)10-s + (0.994 − 0.108i)11-s + (−0.161 − 0.986i)12-s + (−0.907 − 0.419i)13-s + (−0.267 − 0.963i)14-s + (0.0541 + 0.998i)15-s + (−0.994 − 0.108i)16-s + (0.468 + 0.883i)17-s + ⋯ |
L(s) = 1 | + (0.725 − 0.687i)2-s + (0.976 − 0.214i)3-s + (0.0541 − 0.998i)4-s + (−0.161 + 0.986i)5-s + (0.561 − 0.827i)6-s + (0.468 − 0.883i)7-s + (−0.647 − 0.762i)8-s + (0.907 − 0.419i)9-s + (0.561 + 0.827i)10-s + (0.994 − 0.108i)11-s + (−0.161 − 0.986i)12-s + (−0.907 − 0.419i)13-s + (−0.267 − 0.963i)14-s + (0.0541 + 0.998i)15-s + (−0.994 − 0.108i)16-s + (0.468 + 0.883i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.491784367 - 1.824279621i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.491784367 - 1.824279621i\) |
\(L(1)\) |
\(\approx\) |
\(1.866652368 - 0.9098685522i\) |
\(L(1)\) |
\(\approx\) |
\(1.866652368 - 0.9098685522i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 59 | \( 1 \) |
good | 2 | \( 1 + (0.725 - 0.687i)T \) |
| 3 | \( 1 + (0.976 - 0.214i)T \) |
| 5 | \( 1 + (-0.161 + 0.986i)T \) |
| 7 | \( 1 + (0.468 - 0.883i)T \) |
| 11 | \( 1 + (0.994 - 0.108i)T \) |
| 13 | \( 1 + (-0.907 - 0.419i)T \) |
| 17 | \( 1 + (0.468 + 0.883i)T \) |
| 19 | \( 1 + (-0.856 + 0.515i)T \) |
| 23 | \( 1 + (0.370 + 0.928i)T \) |
| 29 | \( 1 + (-0.725 - 0.687i)T \) |
| 31 | \( 1 + (0.856 + 0.515i)T \) |
| 37 | \( 1 + (-0.647 + 0.762i)T \) |
| 41 | \( 1 + (-0.370 + 0.928i)T \) |
| 43 | \( 1 + (0.994 + 0.108i)T \) |
| 47 | \( 1 + (0.161 + 0.986i)T \) |
| 53 | \( 1 + (-0.561 + 0.827i)T \) |
| 61 | \( 1 + (0.725 - 0.687i)T \) |
| 67 | \( 1 + (-0.647 - 0.762i)T \) |
| 71 | \( 1 + (-0.161 - 0.986i)T \) |
| 73 | \( 1 + (-0.267 - 0.963i)T \) |
| 79 | \( 1 + (0.976 + 0.214i)T \) |
| 83 | \( 1 + (-0.796 - 0.605i)T \) |
| 89 | \( 1 + (0.725 + 0.687i)T \) |
| 97 | \( 1 + (-0.267 + 0.963i)T \) |
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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
−32.3430046399430264540868006790, −31.77717435513124302458784334667, −30.94483125914528319395930022573, −29.745193424486556185739839949242, −27.89640817041230508380202927950, −26.965854583430132595489318156778, −25.53405807224684594495916747730, −24.6993185597097385905963823771, −24.153380185219213958396168192194, −22.359058572813523714036853571224, −21.299084850162274333962953349291, −20.45413075949569233722986880155, −19.08820951812190086852002688846, −17.28133846736478968410322917501, −16.15639613595977496230172862084, −15.00040326116969592675489023668, −14.19906470866435276364076423183, −12.77494475957610797809233255717, −11.84378115804999157965923673625, −9.225325982233792702235976418831, −8.529528425013042958640315483785, −7.12028904535717930439022147625, −5.16769831829409609482848245929, −4.147764217249254539370128773090, −2.32680105920542059120146950219,
1.60693535295582980928122261665, 3.1956345512451963673833561902, 4.22394489429208307433994167439, 6.46913693155575063227031735781, 7.77132446483562033395164298409, 9.715176392977078085382622490515, 10.751690703697532045625626329470, 12.18386288297102932146589227048, 13.60524510810699415895104152529, 14.52598992916204698918125319551, 15.127083715145499381383924620071, 17.41705448296808735807595283262, 19.09906619538008720348381605123, 19.55055768061221666365458704394, 20.80913186344954815390812433607, 21.874463700898203171766482339127, 23.10433607350085811025398230530, 24.17038922113008151687289235494, 25.38442800469309364409328950572, 26.81261473538932240448959863105, 27.56342462772354489070994372164, 29.76895554973894482678945180122, 29.92493109197242463056653512416, 30.92623203567064744851365827310, 32.05326923811800145649235451808