L(s) = 1 | + (0.173 + 0.984i)2-s + (0.866 − 0.5i)3-s + (−0.939 + 0.342i)4-s + (−0.573 − 0.819i)5-s + (0.642 + 0.766i)6-s + (−0.965 − 0.258i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (0.707 − 0.707i)10-s + (0.819 − 0.573i)11-s + (−0.642 + 0.766i)12-s + (−0.996 − 0.0871i)13-s + (0.0871 − 0.996i)14-s + (−0.906 − 0.422i)15-s + (0.766 − 0.642i)16-s + (−0.258 − 0.965i)17-s + ⋯ |
L(s) = 1 | + (0.173 + 0.984i)2-s + (0.866 − 0.5i)3-s + (−0.939 + 0.342i)4-s + (−0.573 − 0.819i)5-s + (0.642 + 0.766i)6-s + (−0.965 − 0.258i)7-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (0.707 − 0.707i)10-s + (0.819 − 0.573i)11-s + (−0.642 + 0.766i)12-s + (−0.996 − 0.0871i)13-s + (0.0871 − 0.996i)14-s + (−0.906 − 0.422i)15-s + (0.766 − 0.642i)16-s + (−0.258 − 0.965i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.018425782 - 0.7417373369i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.018425782 - 0.7417373369i\) |
\(L(1)\) |
\(\approx\) |
\(1.053855763 - 0.05262985043i\) |
\(L(1)\) |
\(\approx\) |
\(1.053855763 - 0.05262985043i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 73 | \( 1 \) |
good | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (-0.573 - 0.819i)T \) |
| 7 | \( 1 + (-0.965 - 0.258i)T \) |
| 11 | \( 1 + (0.819 - 0.573i)T \) |
| 13 | \( 1 + (-0.996 - 0.0871i)T \) |
| 17 | \( 1 + (-0.258 - 0.965i)T \) |
| 19 | \( 1 + (-0.984 + 0.173i)T \) |
| 23 | \( 1 + (0.984 + 0.173i)T \) |
| 29 | \( 1 + (0.573 - 0.819i)T \) |
| 31 | \( 1 + (0.422 + 0.906i)T \) |
| 37 | \( 1 + (0.173 - 0.984i)T \) |
| 41 | \( 1 + (-0.766 - 0.642i)T \) |
| 43 | \( 1 + (-0.258 + 0.965i)T \) |
| 47 | \( 1 + (0.0871 + 0.996i)T \) |
| 53 | \( 1 + (-0.819 - 0.573i)T \) |
| 59 | \( 1 + (-0.0871 + 0.996i)T \) |
| 61 | \( 1 + (0.642 - 0.766i)T \) |
| 67 | \( 1 + (-0.642 - 0.766i)T \) |
| 71 | \( 1 + (-0.173 - 0.984i)T \) |
| 79 | \( 1 + (0.642 + 0.766i)T \) |
| 83 | \( 1 + (0.707 - 0.707i)T \) |
| 89 | \( 1 + (0.766 - 0.642i)T \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
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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
−31.38076280857478829913398385153, −30.519480954404581840827114655728, −29.59551637936887182412705201605, −28.182788415715288688113466614800, −27.193149052986666470966643298263, −26.3583846632106267050307587173, −25.27893539943058023280355481605, −23.51319207252403116068705758184, −22.21933200021484728448133522701, −21.8407401323169829655819021375, −20.22722669395314550850061202809, −19.40331547282929218617827376862, −18.88735740204825975673944927837, −17.06129207137279562284848557148, −15.17251466694397726713645036557, −14.697354348398368219382027549258, −13.21844174317497687292704144214, −12.0767239881532685684634007350, −10.58733058742408519884476797028, −9.71072174323837357605798223092, −8.53127104735048636275305810386, −6.75519038814665091980593455359, −4.49908614315020668466944319910, −3.41013232658248460650450660179, −2.310318407398320587138362236789,
0.53636920448869303135842691074, 3.2652728708267837707687532937, 4.56072734338215920854184320305, 6.45713693516702205543914145696, 7.48783781424689271237847834666, 8.734969184969668204113252654804, 9.52772439723326906551155055382, 12.178127332629370406780707356603, 13.04875509407421406867221678588, 14.09986754497979825599930769109, 15.306933905865385175194247300640, 16.35560059082870517630991868908, 17.389709424486275157942034273743, 19.12323244828661542522269512316, 19.628120680955905852415134757400, 21.17333590653261276490029562425, 22.66714924214774024366255458834, 23.67971196941805217918946944314, 24.77965021235887952512728430955, 25.23461774460834987083325512862, 26.702614566941194211086392742187, 27.27008587520759819256511016441, 29.03547552253841732195296269639, 30.24100900979572264348480573323, 31.60875466138675233177102378031