L(s) = 1 | + (−1.19 − 2.06i)2-s + (−1.84 + 3.20i)4-s + (1.46 − 2.52i)5-s + 4.05·8-s − 6.97·10-s + (−0.676 − 1.17i)11-s + (0.733 − 1.26i)13-s + (−1.13 − 1.96i)16-s − 3.31·17-s − 2.20·19-s + (5.39 + 9.35i)20-s + (−1.61 + 2.79i)22-s + (1.31 − 2.27i)23-s + (−1.76 − 3.05i)25-s − 3.49·26-s + ⋯ |
L(s) = 1 | + (−0.843 − 1.46i)2-s + (−0.924 + 1.60i)4-s + (0.653 − 1.13i)5-s + 1.43·8-s − 2.20·10-s + (−0.204 − 0.353i)11-s + (0.203 − 0.352i)13-s + (−0.284 − 0.492i)16-s − 0.802·17-s − 0.506·19-s + (1.20 + 2.09i)20-s + (−0.344 + 0.596i)22-s + (0.274 − 0.474i)23-s + (−0.353 − 0.611i)25-s − 0.686·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5721915741\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5721915741\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.19 + 2.06i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.46 + 2.52i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.676 + 1.17i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.733 + 1.26i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3.31T + 17T^{2} \) |
| 19 | \( 1 + 2.20T + 19T^{2} \) |
| 23 | \( 1 + (-1.31 + 2.27i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.521 + 0.903i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.63 + 2.83i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + (0.904 - 1.56i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.17 + 3.76i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.98 + 3.44i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 6.45T + 53T^{2} \) |
| 59 | \( 1 + (-6.10 + 10.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.279 - 0.484i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.40 - 11.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + (0.383 + 0.664i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.983 + 1.70i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6.40T + 89T^{2} \) |
| 97 | \( 1 + (-4.14 - 7.17i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.046796227466455896948800613164, −8.694619936572947928736231792116, −8.007743282996066949668649393609, −6.62782478933207611414408113381, −5.50568515531254425849008258991, −4.58830082777263733040026613514, −3.53660210285634723103077047702, −2.38473912817936160507814042443, −1.50306072337415781144239006228, −0.31204070263412432918854034009,
1.77990857596007214772214979914, 3.10012791874377513048963430679, 4.60960352772320597966327625306, 5.59350081695350896790500670038, 6.44781520396583281612011045964, 6.85704723561580915235111582522, 7.57277968066438950226485529654, 8.578975123972315137040873803950, 9.165324977343064058011717914342, 10.05299285763376047242890427327