L(s) = 1 | + (−0.5 + 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + 7-s + 8-s + 9-s + 10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + 17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + 7-s + 8-s + 9-s + 10-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + 17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9410416867 + 0.1824016528i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9410416867 + 0.1824016528i\) |
\(L(1)\) |
\(\approx\) |
\(1.007438692 + 0.1993754872i\) |
\(L(1)\) |
\(\approx\) |
\(1.007438692 + 0.1993754872i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 73 | \( 1 \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 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 | \( 1 + (-0.5 + 0.866i)T \) |
| 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
−31.15553797638683711770299687269, −30.34051292645008162247575215710, −29.789631235959694182854531172043, −27.85257746555381096560850328700, −27.32813582931822567816888149137, −26.17853168183116024312716775420, −25.46846712415782882622518601743, −23.82936169736425687330353899379, −22.432841731337003076643078146322, −21.239305605646378376944332655559, −20.35500109715120011705491412705, −19.3899880735064161850010639041, −18.3881436022799132407780897818, −17.47684029479902530421887497852, −15.43453553254422151351743447951, −14.560117871655700864455908899779, −13.21258377790860175032007162889, −11.89419632608690371449419062097, −10.60839548784293371459828935510, −9.625831656541843497849012177422, −7.92906945612528956194629492225, −7.57097805563327699201604114370, −4.577370201522971848124573512, −3.17766410448070010016995199459, −2.01668485661567767382034843840,
1.60642668670275290712781566834, 4.08553503783596115196254133370, 5.35327045338311684881736376023, 7.3883683645836790468019409348, 8.28294768707544917572032769288, 9.011646037015392759168497725, 10.55088199714902184236459744746, 12.4135206552701795225515779001, 13.990530248514427279043360432886, 14.65541257912965393705691196270, 16.00632723097284044591235070469, 16.80523466507783422931016324695, 18.43347361306306250176439361733, 19.240476931000344304768758917366, 20.46202328816979213717894676632, 21.4561363789269550684514257228, 23.55935337006795795017268314164, 24.21285121942538884597019088393, 24.9808064378093677868592759805, 26.262455300384007488585696822612, 27.152368275994982443104124864920, 27.875809308446503237819739950901, 29.35981542609931439095372345416, 31.03663066977544343650550104839, 31.75763361427223077483613646744