L(s) = 1 | + (0.766 + 0.642i)2-s + (−0.5 + 0.866i)3-s + (0.173 + 0.984i)4-s + (0.766 − 0.642i)5-s + (−0.939 + 0.342i)6-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + 10-s + (0.766 − 0.642i)11-s + (−0.939 − 0.342i)12-s + (−0.939 + 0.342i)13-s + (−0.939 + 0.342i)14-s + (0.173 + 0.984i)15-s + (−0.939 + 0.342i)16-s + (−0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)2-s + (−0.5 + 0.866i)3-s + (0.173 + 0.984i)4-s + (0.766 − 0.642i)5-s + (−0.939 + 0.342i)6-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s + 10-s + (0.766 − 0.642i)11-s + (−0.939 − 0.342i)12-s + (−0.939 + 0.342i)13-s + (−0.939 + 0.342i)14-s + (0.173 + 0.984i)15-s + (−0.939 + 0.342i)16-s + (−0.5 − 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7870351753 + 0.9587100501i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7870351753 + 0.9587100501i\) |
\(L(1)\) |
\(\approx\) |
\(1.067595500 + 0.7692809331i\) |
\(L(1)\) |
\(\approx\) |
\(1.067595500 + 0.7692809331i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 73 | \( 1 \) |
good | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.766 - 0.642i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.766 - 0.642i)T \) |
| 13 | \( 1 + (-0.939 + 0.342i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.766 + 0.642i)T \) |
| 23 | \( 1 + (0.766 - 0.642i)T \) |
| 29 | \( 1 + (0.766 + 0.642i)T \) |
| 31 | \( 1 + (0.173 - 0.984i)T \) |
| 37 | \( 1 + (0.766 - 0.642i)T \) |
| 41 | \( 1 + (-0.939 - 0.342i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.939 - 0.342i)T \) |
| 53 | \( 1 + (0.766 + 0.642i)T \) |
| 59 | \( 1 + (-0.939 + 0.342i)T \) |
| 61 | \( 1 + (-0.939 - 0.342i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.766 + 0.642i)T \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.939 + 0.342i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−30.64809364293444331610314361225, −30.29115860544085422548638148658, −29.20381673325601713006623324424, −28.695371157020855031206756677597, −27.08915553439617980914014976661, −25.49649581511352283741743047447, −24.52452294281426015733505121620, −23.30984529719498864364332953441, −22.505081336849408529383705838618, −21.725930698941975279892234730966, −19.96578948620641239168248564261, −19.36807520726128312719233247060, −17.90571105400558463325363947288, −17.03962700214038601524369584313, −15.053842158491011455355604051148, −13.84299698856010255197387260135, −13.11451802107246129859082813560, −11.9196905910505250189908175430, −10.68320734841135990119582733920, −9.69935154727712667292774121087, −7.10156078191428579224024508770, −6.390040695925734562791770819165, −4.8942945051503390342885588419, −3.00188476023841031011816118727, −1.52170726701077121631606929227,
2.89718301436746889267951258882, 4.59261606799748548642688590668, 5.569683982265199917163640197229, 6.56758196426818504564923803792, 8.77403742266912525147272623016, 9.61251127120873094582369183243, 11.58930802277074777433937964953, 12.472746559786627792519944916616, 13.9234716567792415904085485143, 15.02359351472443731723110151139, 16.32841480837927979786480238911, 16.77811422552402282279108921907, 18.100607630435774991134222386960, 20.15765886149020228275675331909, 21.386076776190404806948418584690, 21.993041050696727607972033561792, 22.83200182638407133294759682790, 24.472640647992298409621774027414, 25.00898868099110453727047432795, 26.36762205974479587194216348743, 27.38546814851673303347077098756, 28.89354279830233186261282902601, 29.41902871106197625708013455451, 31.32505971263098151796212269370, 32.0799140472923617432919492606