L(s) = 1 | + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (0.939 − 0.342i)5-s + 7-s + (−0.5 + 0.866i)8-s + (−0.766 + 0.642i)10-s + (0.5 + 0.866i)11-s + (−0.173 − 0.984i)13-s + (−0.939 + 0.342i)14-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + (0.5 − 0.866i)20-s + (−0.766 − 0.642i)22-s + (−0.766 + 0.642i)23-s + (0.766 − 0.642i)25-s + (0.5 + 0.866i)26-s + ⋯ |
L(s) = 1 | + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (0.939 − 0.342i)5-s + 7-s + (−0.5 + 0.866i)8-s + (−0.766 + 0.642i)10-s + (0.5 + 0.866i)11-s + (−0.173 − 0.984i)13-s + (−0.939 + 0.342i)14-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + (0.5 − 0.866i)20-s + (−0.766 − 0.642i)22-s + (−0.766 + 0.642i)23-s + (0.766 − 0.642i)25-s + (0.5 + 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9645633776 + 0.02611032014i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9645633776 + 0.02611032014i\) |
\(L(1)\) |
\(\approx\) |
\(0.9017979350 + 0.04720519401i\) |
\(L(1)\) |
\(\approx\) |
\(0.9017979350 + 0.04720519401i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 5 | \( 1 + (0.939 - 0.342i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.173 - 0.984i)T \) |
| 17 | \( 1 + (-0.766 - 0.642i)T \) |
| 23 | \( 1 + (-0.766 + 0.642i)T \) |
| 29 | \( 1 + (0.173 + 0.984i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 + (0.766 + 0.642i)T \) |
| 47 | \( 1 + (-0.173 - 0.984i)T \) |
| 53 | \( 1 + (0.173 + 0.984i)T \) |
| 59 | \( 1 + (0.173 - 0.984i)T \) |
| 61 | \( 1 + (-0.939 - 0.342i)T \) |
| 67 | \( 1 + (0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 + (-0.939 + 0.342i)T \) |
| 79 | \( 1 + (-0.173 + 0.984i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (-0.939 - 0.342i)T \) |
| 97 | \( 1 + (0.939 - 0.342i)T \) |
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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
−27.45990177017637137890509729995, −26.564831736265773878031788901, −25.93494477046579325616708951299, −24.57616050291130526777823626264, −24.25771581905216359644163419502, −22.29303997853143693390016011268, −21.42757811527983394406189260719, −20.88504637982777376695225801042, −19.53216850920036907477761649656, −18.69665215767792322129559304032, −17.640066917677546704316368303681, −17.14051218867630962447165243170, −15.94814234922821032396382723055, −14.53238005162911695235443048771, −13.68551254527397364359452434201, −12.15767741134650754289937905857, −11.15987362279123016182323974111, −10.36543042597911259240473346492, −9.118910263601267700290637407736, −8.363947022071212676875766018647, −6.93135877283636379993179665851, −5.95908169659581398564656471352, −4.155823877055796728438957008074, −2.46332485554940538922236859155, −1.4983454780685681266558948728,
1.34698723395357512679822850544, 2.40212875993002881703497692203, 4.76492468028576658864181771394, 5.7704097300355757699481508227, 7.054877369532321686391339134116, 8.14459283227473112591831741170, 9.22243340528333786710773017551, 10.08083862900809954325397793927, 11.166576106123841633917046127983, 12.36601437014854864395561947855, 13.846683515942523785007493904688, 14.79060410117791347280653200377, 15.78577709158802824365159591727, 17.13095468331928039872041988891, 17.68092442594150842979367571214, 18.3001159715422847235124318307, 19.95282171805460609248835910509, 20.43080174324087539890619809785, 21.52669594628277019743292732040, 22.80495174609184705403520580296, 24.19501348577161177057293974238, 24.79522362444173900091453699732, 25.55085213580398890372994676181, 26.57317517828528472569344844620, 27.79658730115915953281526587379