L(s) = 1 | + (0.642 − 0.766i)5-s + (0.642 + 0.766i)11-s + (−0.342 − 0.939i)13-s + 17-s − i·19-s + (−0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (0.342 − 0.939i)29-s + (−0.173 + 0.984i)31-s + (−0.866 + 0.5i)37-s + (0.939 − 0.342i)41-s + (−0.984 + 0.173i)43-s + (−0.173 − 0.984i)47-s + (−0.866 + 0.5i)53-s + 55-s + ⋯ |
L(s) = 1 | + (0.642 − 0.766i)5-s + (0.642 + 0.766i)11-s + (−0.342 − 0.939i)13-s + 17-s − i·19-s + (−0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (0.342 − 0.939i)29-s + (−0.173 + 0.984i)31-s + (−0.866 + 0.5i)37-s + (0.939 − 0.342i)41-s + (−0.984 + 0.173i)43-s + (−0.173 − 0.984i)47-s + (−0.866 + 0.5i)53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.08263029953 - 0.6511400698i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.08263029953 - 0.6511400698i\) |
\(L(1)\) |
\(\approx\) |
\(1.051689931 - 0.2411147717i\) |
\(L(1)\) |
\(\approx\) |
\(1.051689931 - 0.2411147717i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.642 - 0.766i)T \) |
| 11 | \( 1 + (0.642 + 0.766i)T \) |
| 13 | \( 1 + (-0.342 - 0.939i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + (-0.939 + 0.342i)T \) |
| 29 | \( 1 + (0.342 - 0.939i)T \) |
| 31 | \( 1 + (-0.173 + 0.984i)T \) |
| 37 | \( 1 + (-0.866 + 0.5i)T \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 + (-0.984 + 0.173i)T \) |
| 47 | \( 1 + (-0.173 - 0.984i)T \) |
| 53 | \( 1 + (-0.866 + 0.5i)T \) |
| 59 | \( 1 + (-0.342 - 0.939i)T \) |
| 61 | \( 1 + (-0.984 + 0.173i)T \) |
| 67 | \( 1 + (0.642 - 0.766i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + (-0.766 + 0.642i)T \) |
| 83 | \( 1 + (-0.342 + 0.939i)T \) |
| 89 | \( 1 - T \) |
| 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
−19.09236076056855719789562712595, −18.61330021873340742204635862685, −17.98119057061078540954949836357, −17.02151619953717982704752166659, −16.60719192462601311085914027434, −15.89006383555908574794215310708, −14.74817925667839075513151565840, −14.19436998532860229887048619881, −14.051803152337533278030614811483, −12.92416177737345069654573036708, −12.1118476248258065377010160417, −11.484569768345927551710862569844, −10.695754715347020522012266110946, −9.965456101620041235786199879024, −9.41139097122836334220688545124, −8.54704317681158759917901097855, −7.66758300688853060588681648530, −6.916282352648636631894621210845, −6.074761863281273328065613346258, −5.731412429852222930348037512691, −4.53576161930996336717981794595, −3.647248026508330419113285613987, −2.99949167930172721343026171036, −1.94053245483018399335439046261, −1.3148327527371411680305437404,
0.09756395942199581221514474104, 1.09860642519869917518996272366, 1.84632843592784506524963377894, 2.78508078065413581109417288399, 3.74726608513607334229226651582, 4.72990093881826633954747089333, 5.24616730412129446620242067668, 6.068616690725506412224539387211, 6.8743344081935668426729941254, 7.78240150168047344229067451411, 8.438291871447144628450551301739, 9.35008344346133591894667416793, 9.85999224411825666645010050821, 10.46080822933838807280107253947, 11.58148876676673529853326111997, 12.33848943317175829050544646444, 12.6737047038878971504837741231, 13.6891117862999786191936020999, 14.1193626449543322366540270751, 15.08835303285883007601038652420, 15.669576446472598921785735576547, 16.50667268616168150812392221225, 17.304576514488433447349587785392, 17.57481543900134700393763938243, 18.329921963597721573761441017059