L(s) = 1 | + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + 13-s − 15-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s + 29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)33-s + (−0.5 − 0.866i)37-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + 13-s − 15-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s + 29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)33-s + (−0.5 − 0.866i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.061967590 - 0.9967111576i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.061967590 - 0.9967111576i\) |
\(L(1)\) |
\(\approx\) |
\(1.063864181 - 0.5273864705i\) |
\(L(1)\) |
\(\approx\) |
\(1.063864181 - 0.5273864705i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 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
−38.012056880405991369357370134, −36.43211096391425170139995700551, −34.96326685022600165474384514866, −33.61549110402238297441508381975, −32.704472805597450589035005577067, −31.1260890119860589331427840442, −30.45265048336728424601549629085, −28.44894235023077189071065115139, −27.232290245779806271458510157928, −26.262991264143702648634788303295, −25.1122673083688761595808017840, −23.08234097223619139240865730245, −22.15369201955020997520079824129, −20.63647515783442067335218092056, −19.50748006991314000785839278723, −17.9478437550643601666056447088, −16.04534733587020059955048832507, −15.056863961563134658233346206581, −13.80075105541795513753832990302, −11.55328266174774739909001050851, −10.26436547748135958326452301573, −8.713323158777394788543792088764, −6.915016985178494680949819668177, −4.52645948358687546331705549285, −2.94219705432514001926433052879,
1.20418351389115677729564000182, 3.67950794765112727902863483477, 6.06990601579822666252741541118, 7.95856753893636974200445830183, 8.97652629868420324973907632654, 11.44929871200261681719221922040, 12.785205470326025408407458484945, 13.9701381300463337575512171827, 15.72570369010992303948758033392, 17.25063118957620484497240638305, 18.86218370939112193562322334688, 19.86647150518276420233711384665, 21.12688779466396732603496086942, 23.17898693515451491262567120763, 24.23089777903988797481216845629, 25.20558753545679701253972511291, 26.730937700287962518124878042004, 28.21598345826257101871560811332, 29.47784821121226517950153682328, 30.798286354285592283427176207998, 31.75724361571701305472450114762, 32.98148065530048461482165625489, 35.05794896882697380710761176655, 35.52027361131090869226211449370, 36.86512116938266328637593646975