L(s) = 1 | + (0.0475 + 0.998i)2-s + (0.723 + 0.690i)3-s + (−0.995 + 0.0950i)4-s + (0.981 − 0.189i)5-s + (−0.654 + 0.755i)6-s + (−0.142 − 0.989i)8-s + (0.0475 + 0.998i)9-s + (0.235 + 0.971i)10-s + (−0.327 + 0.945i)11-s + (−0.786 − 0.618i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)15-s + (0.981 − 0.189i)16-s + (0.580 − 0.814i)17-s + (−0.995 + 0.0950i)18-s + (−0.888 + 0.458i)19-s + ⋯ |
L(s) = 1 | + (0.0475 + 0.998i)2-s + (0.723 + 0.690i)3-s + (−0.995 + 0.0950i)4-s + (0.981 − 0.189i)5-s + (−0.654 + 0.755i)6-s + (−0.142 − 0.989i)8-s + (0.0475 + 0.998i)9-s + (0.235 + 0.971i)10-s + (−0.327 + 0.945i)11-s + (−0.786 − 0.618i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)15-s + (0.981 − 0.189i)16-s + (0.580 − 0.814i)17-s + (−0.995 + 0.0950i)18-s + (−0.888 + 0.458i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4837117818 + 1.691141702i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4837117818 + 1.691141702i\) |
\(L(1)\) |
\(\approx\) |
\(0.9397521524 + 1.016878399i\) |
\(L(1)\) |
\(\approx\) |
\(0.9397521524 + 1.016878399i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 67 | \( 1 \) |
good | 2 | \( 1 + (0.0475 + 0.998i)T \) |
| 3 | \( 1 + (0.723 + 0.690i)T \) |
| 5 | \( 1 + (0.981 - 0.189i)T \) |
| 11 | \( 1 + (-0.327 + 0.945i)T \) |
| 13 | \( 1 + (-0.142 + 0.989i)T \) |
| 17 | \( 1 + (0.580 - 0.814i)T \) |
| 19 | \( 1 + (-0.888 + 0.458i)T \) |
| 23 | \( 1 + (0.235 - 0.971i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (-0.786 + 0.618i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.415 + 0.909i)T \) |
| 43 | \( 1 + (0.415 + 0.909i)T \) |
| 47 | \( 1 + (0.235 - 0.971i)T \) |
| 53 | \( 1 + (-0.995 + 0.0950i)T \) |
| 59 | \( 1 + (-0.786 + 0.618i)T \) |
| 61 | \( 1 + (-0.327 - 0.945i)T \) |
| 71 | \( 1 + (0.415 - 0.909i)T \) |
| 73 | \( 1 + (0.981 + 0.189i)T \) |
| 79 | \( 1 + (-0.786 - 0.618i)T \) |
| 83 | \( 1 + (-0.654 - 0.755i)T \) |
| 89 | \( 1 + (0.723 - 0.690i)T \) |
| 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
−23.55569996410340516152472423468, −22.47021264402798194352575419228, −21.436925484121621617216537477709, −21.10196364936772562640255411503, −20.049911499688167992924641774145, −19.22615536277575063098988617037, −18.66479763324235341311311112478, −17.64589744827148414680292587090, −17.21519279529007952658601530033, −15.42837431913211124530598069731, −14.39317777799686640117741414682, −13.79889547845911622613500754011, −12.94129955024991957585382689106, −12.4894080784076129347911652968, −11.04578391695219199613472934792, −10.3461716120636931896739154786, −9.31070870738733063801068327103, −8.563033865028252440831239886010, −7.61611614028929739027632532429, −6.12679897185731577708700663570, −5.39082158170876015040342120081, −3.7434026971072765331744158679, −2.88148404794511375672205283901, −2.052414275156179876998406726892, −0.923347101156688210281399466425,
1.78640125271318644509567630003, 3.0301621819174564244776452135, 4.54580124816245410733349636204, 4.88285188382705331633457868547, 6.20606574437811901817507783482, 7.15127627947392081656557015863, 8.2577989325862461679550689699, 9.11787580468278958344326876438, 9.777411247386614188163930828176, 10.5176796830278436442776217847, 12.3456479804425235693622132747, 13.201766060993198576120902064912, 14.25278306016991353857329708436, 14.47423831623543070538978937962, 15.58048325561370590802163535966, 16.493037646267472238518530345486, 17.01135807993746617985809305325, 18.14724464591885349597422000658, 18.86114523993851582204142227511, 20.0900446949579329113773184176, 21.12239042150518040281442181635, 21.52911279929516223987213051764, 22.57011926659486385622699784399, 23.36715368396061180318176711388, 24.5391014230036968116762896046