L(s) = 1 | + (0.999 + 0.0380i)2-s + (0.669 − 0.743i)3-s + (0.997 + 0.0760i)4-s + (0.696 − 0.717i)6-s + (0.993 + 0.113i)8-s + (−0.104 − 0.994i)9-s + (0.723 − 0.690i)12-s + (−0.941 + 0.336i)13-s + (0.988 + 0.151i)16-s + (0.683 + 0.730i)17-s + (−0.0665 − 0.997i)18-s + (0.483 + 0.875i)19-s + (0.888 − 0.458i)23-s + (0.749 − 0.662i)24-s + (−0.953 + 0.299i)26-s + (−0.809 − 0.587i)27-s + ⋯ |
L(s) = 1 | + (0.999 + 0.0380i)2-s + (0.669 − 0.743i)3-s + (0.997 + 0.0760i)4-s + (0.696 − 0.717i)6-s + (0.993 + 0.113i)8-s + (−0.104 − 0.994i)9-s + (0.723 − 0.690i)12-s + (−0.941 + 0.336i)13-s + (0.988 + 0.151i)16-s + (0.683 + 0.730i)17-s + (−0.0665 − 0.997i)18-s + (0.483 + 0.875i)19-s + (0.888 − 0.458i)23-s + (0.749 − 0.662i)24-s + (−0.953 + 0.299i)26-s + (−0.809 − 0.587i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(4.877650220 - 1.292369196i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.877650220 - 1.292369196i\) |
\(L(1)\) |
\(\approx\) |
\(2.550990706 - 0.4781475833i\) |
\(L(1)\) |
\(\approx\) |
\(2.550990706 - 0.4781475833i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.999 + 0.0380i)T \) |
| 3 | \( 1 + (0.669 - 0.743i)T \) |
| 13 | \( 1 + (-0.941 + 0.336i)T \) |
| 17 | \( 1 + (0.683 + 0.730i)T \) |
| 19 | \( 1 + (0.483 + 0.875i)T \) |
| 23 | \( 1 + (0.888 - 0.458i)T \) |
| 29 | \( 1 + (0.921 - 0.389i)T \) |
| 31 | \( 1 + (0.991 + 0.132i)T \) |
| 37 | \( 1 + (0.851 + 0.524i)T \) |
| 41 | \( 1 + (-0.985 + 0.170i)T \) |
| 43 | \( 1 + (0.415 - 0.909i)T \) |
| 47 | \( 1 + (-0.0665 + 0.997i)T \) |
| 53 | \( 1 + (-0.988 + 0.151i)T \) |
| 59 | \( 1 + (-0.345 - 0.938i)T \) |
| 61 | \( 1 + (-0.532 - 0.846i)T \) |
| 67 | \( 1 + (-0.928 + 0.371i)T \) |
| 71 | \( 1 + (0.0855 + 0.996i)T \) |
| 73 | \( 1 + (0.935 + 0.353i)T \) |
| 79 | \( 1 + (-0.861 + 0.508i)T \) |
| 83 | \( 1 + (0.998 + 0.0570i)T \) |
| 89 | \( 1 + (0.327 + 0.945i)T \) |
| 97 | \( 1 + (0.198 + 0.980i)T \) |
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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
−18.62540066495302157996682788452, −17.56172981740256422664493495319, −16.80018075606529369859047431764, −16.18120196470184381109094661524, −15.53823696307898870525935126673, −14.953326532947027611731720954308, −14.456880103517135306449679819811, −13.64909866812470290432740197061, −13.30008196178850638807785786580, −12.289242873431539062862476291942, −11.70894636156630073188923067334, −10.92519958936827376475881918222, −10.21197833435603824552744816776, −9.613422585222594914446341562842, −8.842288764987045407877222738275, −7.73877717646810233711244107852, −7.413571304458079265059977859307, −6.4612449283994251821027025781, −5.42900259078121877901091862488, −4.86533911043723729337610034832, −4.4370045043205021982813001299, −3.16623629280087638117572086396, −3.04123122535071181329315538693, −2.183490317328244457490539904625, −1.0139248734777785288562458841,
1.06068962829939460553021368555, 1.76905340484096745294212042979, 2.7071942308126309309488476649, 3.15487415600848149077403039653, 4.08512374607293418372095021504, 4.82048067432236842916027946437, 5.72502372340518749958309907703, 6.494619424095214580464294377239, 6.978066881524691645828931394410, 7.98108506590629925414887374277, 8.14615325835329692464493003953, 9.437831444745483560275089941870, 10.06004292354799699866330263206, 10.9470664920712115266388827556, 11.97597319557232247030561206932, 12.2245624526764621565918185485, 12.87263006860743083523582397558, 13.67082043144171648240058484252, 14.21349965391867835065587123161, 14.72481282118734657185642954694, 15.324591750571459253470988550272, 16.128690817071471222683716415439, 17.06763736056388221650975315088, 17.38181288335927460256648196102, 18.691076903626207117870967978584