L(s) = 1 | + (−0.215 + 0.976i)2-s + (−0.354 − 0.935i)3-s + (−0.906 − 0.421i)4-s + (−0.354 − 0.935i)5-s + (0.989 − 0.144i)6-s + (0.989 + 0.144i)7-s + (0.607 − 0.794i)8-s + (−0.748 + 0.663i)9-s + (0.989 − 0.144i)10-s + (−0.0724 + 0.997i)12-s + (0.168 + 0.985i)13-s + (−0.354 + 0.935i)14-s + (−0.748 + 0.663i)15-s + (0.644 + 0.764i)16-s + (−0.926 + 0.377i)17-s + (−0.485 − 0.873i)18-s + ⋯ |
L(s) = 1 | + (−0.215 + 0.976i)2-s + (−0.354 − 0.935i)3-s + (−0.906 − 0.421i)4-s + (−0.354 − 0.935i)5-s + (0.989 − 0.144i)6-s + (0.989 + 0.144i)7-s + (0.607 − 0.794i)8-s + (−0.748 + 0.663i)9-s + (0.989 − 0.144i)10-s + (−0.0724 + 0.997i)12-s + (0.168 + 0.985i)13-s + (−0.354 + 0.935i)14-s + (−0.748 + 0.663i)15-s + (0.644 + 0.764i)16-s + (−0.926 + 0.377i)17-s + (−0.485 − 0.873i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8312527358 - 0.3571616349i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8312527358 - 0.3571616349i\) |
\(L(1)\) |
\(\approx\) |
\(0.7043785664 + 0.02868298560i\) |
\(L(1)\) |
\(\approx\) |
\(0.7043785664 + 0.02868298560i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (-0.215 + 0.976i)T \) |
| 3 | \( 1 + (-0.354 - 0.935i)T \) |
| 5 | \( 1 + (-0.354 - 0.935i)T \) |
| 7 | \( 1 + (0.989 + 0.144i)T \) |
| 13 | \( 1 + (0.168 + 0.985i)T \) |
| 17 | \( 1 + (-0.926 + 0.377i)T \) |
| 19 | \( 1 + (-0.926 - 0.377i)T \) |
| 23 | \( 1 + (-0.943 + 0.331i)T \) |
| 29 | \( 1 + (-0.215 - 0.976i)T \) |
| 31 | \( 1 + (0.568 + 0.822i)T \) |
| 37 | \( 1 + (-0.681 + 0.732i)T \) |
| 41 | \( 1 + (0.0724 - 0.997i)T \) |
| 43 | \( 1 + (0.998 - 0.0483i)T \) |
| 47 | \( 1 + (0.995 - 0.0965i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.527 + 0.849i)T \) |
| 61 | \( 1 + (0.809 - 0.587i)T \) |
| 67 | \( 1 + (-0.998 - 0.0483i)T \) |
| 71 | \( 1 + (0.958 - 0.285i)T \) |
| 73 | \( 1 + (-0.309 + 0.951i)T \) |
| 79 | \( 1 + (-0.779 + 0.626i)T \) |
| 83 | \( 1 + (-0.120 - 0.992i)T \) |
| 89 | \( 1 + (0.309 - 0.951i)T \) |
| 97 | \( 1 + (0.885 - 0.464i)T \) |
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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
−20.59943296406501930362648820255, −20.11910346739293558549400562648, −19.23003531386056209575155490111, −18.222814770066966148401250667970, −17.80006124758614746173714963924, −17.17227902005310122511897845897, −16.08932744253595725339436810125, −15.24225610514830110886225620751, −14.53546139750984747536377354746, −13.95093775877507030310311882860, −12.77383360320806021123535205462, −11.87448305986010361615307545465, −11.19445617147850986162678075363, −10.66257176461454733556563023006, −10.23830939196027408562951818233, −9.16534938430881172864901489747, −8.318044324809195379336916410819, −7.649848653601628569904073971939, −6.32346068341878992500039359528, −5.32748546515277537183036181423, −4.38362121672645790747555883733, −3.869389364781597771151767845554, −2.87476710434480247883755563012, −2.05401875278318733020067281566, −0.587410889461196089310672867094,
0.33522574509062066306164962527, 1.41237721413989037582699904809, 2.097387650185691264904459643306, 4.15437638487777687630015400215, 4.60648343544178370594727262669, 5.57515489312149617783327944954, 6.28986705670290172161953288070, 7.18008463241929041728147500673, 7.93071535639360506639312370086, 8.63217488981019259950336623914, 9.0268497999544839430378394806, 10.45835445407935502730180511201, 11.38691195363878315139479769783, 12.11103440011005863043921639035, 12.90458679443300637346420598771, 13.76444048096152397284379427778, 14.18453322048388751272014581960, 15.393612742262387904887549883565, 15.86483525499144750002215300472, 17.09471264580023238338014480845, 17.17812451373341117663410854430, 17.94999173821646500041493386394, 18.89969746847969203830142086180, 19.36299150921335488716419978136, 20.26016072412699658194128597794