| L(s) = 1 | + 0.498i·2-s + 1.75·4-s + (−0.192 − 2.22i)5-s − 4.62i·7-s + 1.86i·8-s + (1.10 − 0.0958i)10-s − 4.13·11-s + 2.39i·13-s + 2.30·14-s + 2.57·16-s − 7.34i·17-s − 19-s + (−0.337 − 3.90i)20-s − 2.06i·22-s + 2.69i·23-s + ⋯ |
| L(s) = 1 | + 0.352i·2-s + 0.875·4-s + (−0.0860 − 0.996i)5-s − 1.74i·7-s + 0.660i·8-s + (0.350 − 0.0303i)10-s − 1.24·11-s + 0.663i·13-s + 0.616·14-s + 0.643·16-s − 1.78i·17-s − 0.229·19-s + (−0.0753 − 0.872i)20-s − 0.439i·22-s + 0.562i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0860 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0860 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14950 - 1.05448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14950 - 1.05448i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.192 + 2.22i)T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 0.498iT - 2T^{2} \) |
| 7 | \( 1 + 4.62iT - 7T^{2} \) |
| 11 | \( 1 + 4.13T + 11T^{2} \) |
| 13 | \( 1 - 2.39iT - 13T^{2} \) |
| 17 | \( 1 + 7.34iT - 17T^{2} \) |
| 23 | \( 1 - 2.69iT - 23T^{2} \) |
| 29 | \( 1 - 0.979T + 29T^{2} \) |
| 31 | \( 1 + 5.10T + 31T^{2} \) |
| 37 | \( 1 + 9.85iT - 37T^{2} \) |
| 41 | \( 1 - 4.52T + 41T^{2} \) |
| 43 | \( 1 - 3.43iT - 43T^{2} \) |
| 47 | \( 1 - 5.09iT - 47T^{2} \) |
| 53 | \( 1 + 3.36iT - 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 6.64T + 61T^{2} \) |
| 67 | \( 1 + 7.28iT - 67T^{2} \) |
| 71 | \( 1 - 7.28T + 71T^{2} \) |
| 73 | \( 1 + 3.82iT - 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 3.92iT - 83T^{2} \) |
| 89 | \( 1 - 2.34T + 89T^{2} \) |
| 97 | \( 1 - 2.34iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.989255617762879206850980671284, −9.173281730524369323337162446701, −7.87377141337556799725667357909, −7.51104880839391072895061013956, −6.77047209199081039152831202589, −5.48137048601800048649521916939, −4.74554905687107645550523816081, −3.63714203499263485669972501938, −2.19663119321528483743556905012, −0.70624675174005692316613364532,
2.10613620588282786341810743423, 2.65172072503438398001605964730, 3.61347982216494750984795664305, 5.39596470772121012478815492006, 6.01051356277780458820915828986, 6.81575654225561320193945010548, 7.980391211038154774212227160712, 8.479006292016367664569322468113, 9.881023777837909217723896131561, 10.49829923747880902903601360282
