L(s) = 1 | + (−0.173 + 0.984i)5-s + (−0.766 + 0.642i)7-s + (0.173 + 0.984i)11-s + (−0.939 + 0.342i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (0.939 + 0.342i)29-s + (−0.766 − 0.642i)31-s + (−0.5 − 0.866i)35-s + (−0.5 + 0.866i)37-s + (0.939 − 0.342i)41-s + (−0.173 − 0.984i)43-s + (0.766 − 0.642i)47-s + ⋯ |
L(s) = 1 | + (−0.173 + 0.984i)5-s + (−0.766 + 0.642i)7-s + (0.173 + 0.984i)11-s + (−0.939 + 0.342i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.766 + 0.642i)23-s + (−0.939 − 0.342i)25-s + (0.939 + 0.342i)29-s + (−0.766 − 0.642i)31-s + (−0.5 − 0.866i)35-s + (−0.5 + 0.866i)37-s + (0.939 − 0.342i)41-s + (−0.173 − 0.984i)43-s + (0.766 − 0.642i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0581 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0581 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6144165308 + 0.5796721002i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6144165308 + 0.5796721002i\) |
\(L(1)\) |
\(\approx\) |
\(0.8485765085 + 0.3245992846i\) |
\(L(1)\) |
\(\approx\) |
\(0.8485765085 + 0.3245992846i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.173 + 0.984i)T \) |
| 7 | \( 1 + (-0.766 + 0.642i)T \) |
| 11 | \( 1 + (0.173 + 0.984i)T \) |
| 13 | \( 1 + (-0.939 + 0.342i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.766 + 0.642i)T \) |
| 29 | \( 1 + (0.939 + 0.342i)T \) |
| 31 | \( 1 + (-0.766 - 0.642i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 + (-0.173 - 0.984i)T \) |
| 47 | \( 1 + (0.766 - 0.642i)T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (0.173 - 0.984i)T \) |
| 61 | \( 1 + (0.766 - 0.642i)T \) |
| 67 | \( 1 + (0.939 - 0.342i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.939 + 0.342i)T \) |
| 83 | \( 1 + (-0.939 - 0.342i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.173 + 0.984i)T \) |
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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
−29.22941020866405602077918933673, −28.45662619515928123176660166150, −27.21435020025855947262478044787, −26.41950946660571439407278317325, −25.09218990198217619660863539041, −24.21012743385602857922655305628, −23.28965532057962935120011576419, −22.069855352228012353552176878540, −21.01695068037910995774286068177, −19.74812633895018016899656151209, −19.313647574195472113815111418031, −17.52477720183252138253358226073, −16.64578219994070678814158664472, −15.85113749283445488369100986517, −14.34464349839000652715587860342, −13.12943638931542478043002888062, −12.363417042999333248437877363624, −10.90797796195453146719216500708, −9.6441961248293219076629207696, −8.54971075229711055829102305696, −7.26094382340804555741870667590, −5.802660529673255096840349047900, −4.468049020835512744481938917969, −3.08429665359403964217736272167, −0.86265631827596824109838692750,
2.307543097953968357765214575752, 3.4959945264182356740835830522, 5.23135287643856390755919429374, 6.69497141687295195217784445133, 7.54036985891803344985700986146, 9.37464280233212473772312566550, 10.12708522661485870992560812692, 11.67713808593340777883141404298, 12.48679266615840150047836627057, 14.05194592634709493336621497035, 14.98504829719536992475813025332, 15.95290271520698656152081528894, 17.309347949423411556307404891787, 18.52076051079686394822511952333, 19.20615447948160405733110941323, 20.40933704459019538260544965569, 21.84548750589014811281973531307, 22.55078829915945513685728898757, 23.40506984486889816438160837881, 25.01420198334561672380798712293, 25.62227984670828132539961204444, 26.79100792882036665530614332113, 27.66037905244055784074541013663, 29.02300079736038385607683762134, 29.65256498854089114210518185174