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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6024857757 + 0.2986686198i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6024857757 + 0.2986686198i\) |
\(L(1)\) |
\(\approx\) |
\(0.8272499164 + 0.2685074801i\) |
\(L(1)\) |
\(\approx\) |
\(0.8272499164 + 0.2685074801i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
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
−36.934593998166979214811617900949, −36.17490121615785099863749800058, −35.03367287706022271367390804396, −33.72917690559028311256085310793, −32.4409782555275931724620137368, −30.97537086303880087875832717093, −29.67465296411982472065265146399, −28.68629501827149143803670306400, −27.59488233323263092361840342523, −25.492121781285813810652627990171, −24.61480591540406694707848574615, −23.41929007401572973809577896965, −22.02477477575038040580831517844, −20.39283486937043947183748069495, −19.05214065294191287469801220805, −17.45111309920348937368431191973, −16.74918897964851249538494892171, −14.53767518302005912557469765011, −12.88497151683190396572405260016, −12.08780839164623111600817757729, −10.035048155259889179243112063613, −8.20025188815811285412341405267, −6.48949348354827902421124253680, −4.90145218572609997721407147630, −1.79104060299590196928243808855,
3.168742768140166422476904919, 5.219666932069924128995417747349, 6.794214378265738185173637197862, 9.23360735363388848724695713493, 10.51423245698153921537833739077, 11.74999424202570888718609457837, 13.96407991421429788322836586322, 15.179005396573225630603026027661, 16.708192258780626926637165157904, 17.87467441006700468299801803899, 19.51938201887740370352982841590, 21.381734511895633760502523152499, 22.06331783005413527696043861485, 23.35211512927444184464274455383, 25.16076152890179732894662818403, 26.59554445936726606150763972526, 27.39004370038808187134093669404, 29.03229979414819473139628004895, 29.924728159971386673695517524619, 31.75307586653671109167324601716, 32.87306776536224552253934478215, 34.06310863748782348826866600632, 34.78617442325068261563198949791, 36.78694257822323976854449062769, 37.90490365720337567833703476823