L(s) = 1 | + (0.669 − 0.743i)2-s + (−0.104 − 0.994i)4-s + (−0.669 − 0.743i)5-s + (−0.913 − 0.406i)7-s + (−0.809 − 0.587i)8-s − 10-s + (0.978 + 0.207i)13-s + (−0.913 + 0.406i)14-s + (−0.978 + 0.207i)16-s + (0.309 − 0.951i)17-s + (0.809 + 0.587i)19-s + (−0.669 + 0.743i)20-s + (0.5 − 0.866i)23-s + (−0.104 + 0.994i)25-s + (0.809 − 0.587i)26-s + ⋯ |
L(s) = 1 | + (0.669 − 0.743i)2-s + (−0.104 − 0.994i)4-s + (−0.669 − 0.743i)5-s + (−0.913 − 0.406i)7-s + (−0.809 − 0.587i)8-s − 10-s + (0.978 + 0.207i)13-s + (−0.913 + 0.406i)14-s + (−0.978 + 0.207i)16-s + (0.309 − 0.951i)17-s + (0.809 + 0.587i)19-s + (−0.669 + 0.743i)20-s + (0.5 − 0.866i)23-s + (−0.104 + 0.994i)25-s + (0.809 − 0.587i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4898288350 - 1.003941666i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4898288350 - 1.003941666i\) |
\(L(1)\) |
\(\approx\) |
\(0.8951499942 - 0.7549105015i\) |
\(L(1)\) |
\(\approx\) |
\(0.8951499942 - 0.7549105015i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.669 - 0.743i)T \) |
| 5 | \( 1 + (-0.669 - 0.743i)T \) |
| 7 | \( 1 + (-0.913 - 0.406i)T \) |
| 13 | \( 1 + (0.978 + 0.207i)T \) |
| 17 | \( 1 + (0.309 - 0.951i)T \) |
| 19 | \( 1 + (0.809 + 0.587i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.913 + 0.406i)T \) |
| 31 | \( 1 + (-0.978 - 0.207i)T \) |
| 37 | \( 1 + (-0.809 + 0.587i)T \) |
| 41 | \( 1 + (0.913 - 0.406i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.104 - 0.994i)T \) |
| 53 | \( 1 + (-0.309 - 0.951i)T \) |
| 59 | \( 1 + (0.104 + 0.994i)T \) |
| 61 | \( 1 + (0.978 - 0.207i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.669 + 0.743i)T \) |
| 83 | \( 1 + (-0.978 + 0.207i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.669 - 0.743i)T \) |
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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
−30.67603908979738174644078400068, −29.649186087661453024516862939485, −28.23671461686415338675797351727, −26.949768784762140239192744791276, −25.96842590645936856406193556554, −25.345391384917036875509573252918, −23.904054042672889622860068313690, −23.077357781854729484085935016291, −22.29348180432868045671015220685, −21.3044914565016576771934451706, −19.77271042478908477122871852882, −18.66108255115476196342711924539, −17.49733562098809440088558264046, −15.9577918670037813891985896367, −15.56624898055811833090337882188, −14.32827400716910882689352829755, −13.15666231771682981015358147191, −12.09603925692418101566009637166, −10.860948794258228686291924251320, −9.10751999010381992592008787730, −7.79285245258613315449835803567, −6.67848907760289786183782860652, −5.65399850340929376851119127377, −3.85862498465423362260759663542, −2.98754885143829296256813423913,
1.01405192946047354218048425028, 3.1338716809577441614531339084, 4.165199817724815838314681026023, 5.50356260432876560961568056291, 6.97800799290622928017844163486, 8.76774218743570895513635285564, 9.91955637089759072668679204305, 11.223036395091245509029632247371, 12.29252509045439489083162132873, 13.175815177105829513390518290844, 14.23792341137872847889778174076, 15.792036704174685518555744717329, 16.41963736464034380487843499523, 18.35120959172219137049002070522, 19.334991396428240323612758470466, 20.31430464206063422593000520296, 20.9687110386731942077013806558, 22.54806040655582100364642676365, 23.12167758219989719435518925511, 24.13467367359769079853858973622, 25.27325702845854947765461517099, 26.81258764382110023162621429293, 27.819566864781486567045906290842, 28.800820948343724387488698442999, 29.507576219833981587368296118823