L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.104 − 0.994i)5-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)10-s + (−0.104 + 0.994i)13-s + (−0.809 + 0.587i)16-s + (−0.104 − 0.994i)17-s + (−0.978 + 0.207i)19-s + (0.913 − 0.406i)20-s + (−0.5 + 0.866i)23-s + (−0.978 + 0.207i)25-s + (0.669 − 0.743i)26-s + (−0.978 − 0.207i)29-s + (−0.809 − 0.587i)31-s + 32-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.104 − 0.994i)5-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)10-s + (−0.104 + 0.994i)13-s + (−0.809 + 0.587i)16-s + (−0.104 − 0.994i)17-s + (−0.978 + 0.207i)19-s + (0.913 − 0.406i)20-s + (−0.5 + 0.866i)23-s + (−0.978 + 0.207i)25-s + (0.669 − 0.743i)26-s + (−0.978 − 0.207i)29-s + (−0.809 − 0.587i)31-s + 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02142973801 - 0.04622234867i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02142973801 - 0.04622234867i\) |
\(L(1)\) |
\(\approx\) |
\(0.5047107137 - 0.1944412239i\) |
\(L(1)\) |
\(\approx\) |
\(0.5047107137 - 0.1944412239i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.104 - 0.994i)T \) |
| 13 | \( 1 + (-0.104 + 0.994i)T \) |
| 17 | \( 1 + (-0.104 - 0.994i)T \) |
| 19 | \( 1 + (-0.978 + 0.207i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.978 - 0.207i)T \) |
| 31 | \( 1 + (-0.809 - 0.587i)T \) |
| 37 | \( 1 + (0.669 - 0.743i)T \) |
| 41 | \( 1 + (-0.978 + 0.207i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.309 - 0.951i)T \) |
| 53 | \( 1 + (0.913 - 0.406i)T \) |
| 59 | \( 1 + (0.309 + 0.951i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (-0.978 - 0.207i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (-0.104 - 0.994i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.913 - 0.406i)T \) |
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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
−23.41378308858110066473674047557, −22.43510185427562613438991127620, −21.76541411442844095795608304800, −20.43077217643413689844069898371, −19.79102607529281502032715035367, −18.85985790329044985829854407944, −18.36220612262397258333769049833, −17.46043164201679030353205074091, −16.83012622559683733907136927998, −15.70846543779924976026574169684, −14.98934074255662225691930169782, −14.56379886140683723748092973097, −13.41198836576124544939199290939, −12.315204592083580207978256978923, −11.01323777525991433804565637845, −10.59527396817973425104643980883, −9.81552973568579413390824458806, −8.60977432773322728626299238051, −7.946672450886037607349778290451, −6.981445793784486830498211356387, −6.26517921446167576541990946417, −5.3917635208305165457670838392, −4.00855672732496144631314475894, −2.74416981100242078467206618016, −1.72430229742353013676523345598,
0.03105219618375294917405317032, 1.478474298500525921161233190632, 2.29438168329973351930268487059, 3.74698043679864426719395340855, 4.46892866244183839997375844099, 5.73783591281375707189918205680, 7.030430383718700721164701943258, 7.826413152446473432481877485074, 8.8450166227476120755201934235, 9.33645181814709210270241518321, 10.220487142954582297138715716462, 11.53403634193746559625253347394, 11.77024388847521951915483979849, 12.952917273625927893772791479622, 13.47451606429902178818510176679, 14.82947186516271652275095550713, 15.98230206527957877498562703699, 16.575578484136822438944764191027, 17.175736448586539244428321360717, 18.19262609288593845067239492879, 18.91752017537161313355849200445, 19.83148962785604639834071398330, 20.33819850277207226985283578029, 21.28408291678960181211903332193, 21.71783327068613071190939114584