L(s) = 1 | + (0.890 + 0.454i)2-s + (0.503 + 0.863i)3-s + (0.586 + 0.809i)4-s + (0.751 − 0.659i)5-s + (0.0558 + 0.998i)6-s + (0.227 + 0.973i)7-s + (0.154 + 0.987i)8-s + (−0.492 + 0.870i)9-s + (0.969 − 0.245i)10-s + (−0.977 − 0.209i)11-s + (−0.404 + 0.914i)12-s + (0.179 + 0.983i)13-s + (−0.239 + 0.970i)14-s + (0.948 + 0.317i)15-s + (−0.311 + 0.950i)16-s + (−0.449 − 0.893i)17-s + ⋯ |
L(s) = 1 | + (0.890 + 0.454i)2-s + (0.503 + 0.863i)3-s + (0.586 + 0.809i)4-s + (0.751 − 0.659i)5-s + (0.0558 + 0.998i)6-s + (0.227 + 0.973i)7-s + (0.154 + 0.987i)8-s + (−0.492 + 0.870i)9-s + (0.969 − 0.245i)10-s + (−0.977 − 0.209i)11-s + (−0.404 + 0.914i)12-s + (0.179 + 0.983i)13-s + (−0.239 + 0.970i)14-s + (0.948 + 0.317i)15-s + (−0.311 + 0.950i)16-s + (−0.449 − 0.893i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.634264039 + 2.487973133i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.634264039 + 2.487973133i\) |
\(L(1)\) |
\(\approx\) |
\(1.728298855 + 1.285046052i\) |
\(L(1)\) |
\(\approx\) |
\(1.728298855 + 1.285046052i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (0.890 + 0.454i)T \) |
| 3 | \( 1 + (0.503 + 0.863i)T \) |
| 5 | \( 1 + (0.751 - 0.659i)T \) |
| 7 | \( 1 + (0.227 + 0.973i)T \) |
| 11 | \( 1 + (-0.977 - 0.209i)T \) |
| 13 | \( 1 + (0.179 + 0.983i)T \) |
| 17 | \( 1 + (-0.449 - 0.893i)T \) |
| 19 | \( 1 + (0.323 - 0.946i)T \) |
| 29 | \( 1 + (0.346 - 0.938i)T \) |
| 31 | \( 1 + (-0.806 + 0.591i)T \) |
| 37 | \( 1 + (0.0310 + 0.999i)T \) |
| 41 | \( 1 + (0.901 - 0.432i)T \) |
| 43 | \( 1 + (0.827 - 0.561i)T \) |
| 47 | \( 1 + (0.203 - 0.979i)T \) |
| 53 | \( 1 + (0.969 + 0.245i)T \) |
| 59 | \( 1 + (-0.0434 - 0.999i)T \) |
| 61 | \( 1 + (-0.287 - 0.957i)T \) |
| 67 | \( 1 + (0.997 - 0.0744i)T \) |
| 71 | \( 1 + (0.524 + 0.851i)T \) |
| 73 | \( 1 + (0.346 + 0.938i)T \) |
| 79 | \( 1 + (0.437 + 0.899i)T \) |
| 83 | \( 1 + (0.545 + 0.837i)T \) |
| 89 | \( 1 + (-0.616 - 0.787i)T \) |
| 97 | \( 1 + (0.299 - 0.954i)T \) |
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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
−23.11715103521141547723539939012, −22.62625122471572897293932077501, −21.380872585429479129151172449083, −20.71463742028204858173092769888, −20.02308937511674897404575134726, −19.198202487484260294866696648259, −18.14060700779285142696793456316, −17.72599426729084142831063272905, −16.31815642292641314249398471437, −15.00022973009905720533823823982, −14.519699919315936090572292206729, −13.628530968641702984003840830976, −13.06550824668951086863648384654, −12.40627666469563660357515689113, −10.87062598425101590115691875212, −10.58189548010021746279845856786, −9.47065306302799251296970068643, −7.88985377971146018879853395971, −7.24749616914119309117670171056, −6.17341285860316247225022639713, −5.47819291542254567867145265717, −3.97506752214457003589578712042, −3.04375339821949669708177795595, −2.15655164102488240266047865774, −1.15970467053703213408878328273,
2.208413404571396083015556814381, 2.69179705432039924083948121279, 4.12741726778305047001564867735, 5.11299847887277605174344571751, 5.431772033992633138219720781309, 6.72260214132524954906289634447, 8.07577491294839336402072372167, 8.85686162333283694566836083668, 9.5420440799837154677955313461, 10.91753933727601648942382986933, 11.7408549977676755847452594847, 12.84813453563399834149725449161, 13.73011844002853828733668909020, 14.186468792309990480718596989931, 15.47943642122719246708953681652, 15.77776620744504488069631265748, 16.63791736507277920995943527167, 17.59603567867501376426739586275, 18.6403734966716007218748483466, 20.01950166643256846382102607962, 20.72947258955422945880781964584, 21.511278185267905410281909662346, 21.707289070126145552798607069079, 22.74383726560862481785205253661, 23.943859454517346928269296128544