L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + 8-s + (0.5 + 0.866i)11-s − 13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s − 22-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)26-s − 29-s + (−0.5 − 0.866i)31-s + (−0.5 − 0.866i)32-s + 34-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + 8-s + (0.5 + 0.866i)11-s − 13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s − 22-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)26-s − 29-s + (−0.5 − 0.866i)31-s + (−0.5 − 0.866i)32-s + 34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.06533879300 + 0.2789141793i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.06533879300 + 0.2789141793i\) |
\(L(1)\) |
\(\approx\) |
\(0.5493771009 + 0.2723412023i\) |
\(L(1)\) |
\(\approx\) |
\(0.5493771009 + 0.2723412023i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 - T \) |
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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
−28.87548302251100609168071667510, −27.93179778206207243352133655515, −26.896822382989606730414675259218, −26.17724397322768005436317674164, −24.84344069285568189236667077100, −23.686225419326703265294688102343, −22.04601782809088400972718170663, −21.79244521856759348986315582559, −20.27544545598883976714070210571, −19.51099632786497457570814517202, −18.53881233997781784872207685699, −17.308381471025647728030947822775, −16.55318097462350856377297785837, −14.89930132395357707934589529509, −13.56924233613953713961145621733, −12.50005525862979035440925920404, −11.38762608512067854001826790455, −10.38503326779080928226001006594, −9.135631861667750098523053841659, −8.16382531935513175056389747299, −6.64928949692937849352736745075, −4.755073300678322130690741390826, −3.38397176177659754836981306601, −1.93424418653199510222221150194, −0.14096964910544635228477804826,
1.88921955558837161199913812121, 4.228694793321656657799976381, 5.49838697343202041384613665660, 6.887456644001115842130507593974, 7.78808413960726810088376298326, 9.264072879993976079549714505794, 10.02416795740906573723465796983, 11.5945825988443723407450962696, 13.05527651198810013912438132846, 14.4088456420389496956251977844, 15.14757933211887186189560277597, 16.41407208990373057305120863751, 17.32750765027055255176422823665, 18.276418771017353473044460801604, 19.449403963525748085958778709056, 20.38409126647060083294580318779, 22.0945345462781017969679419520, 22.93090849834523702737738739310, 24.084120872907276272492329698855, 25.00452398899386431373514057034, 25.82261535683761852748668378458, 27.0192396528579788513391888038, 27.687660240021414207853854844120, 28.829048007714046958839570566335, 29.89868223375412927209055191513