| L(s) = 1 | + (−0.288 + 0.128i)2-s + (−1.27 + 1.41i)4-s + (3.70 + 1.64i)5-s + (−3.33 − 0.709i)7-s + (0.380 − 1.17i)8-s − 1.28·10-s + (−3.27 − 0.553i)11-s + (−0.144 + 1.37i)13-s + (1.05 − 0.224i)14-s + (−0.356 − 3.39i)16-s + (−5.20 + 3.78i)17-s + (−1.07 + 3.31i)19-s + (−7.03 + 3.13i)20-s + (1.01 − 0.260i)22-s + (−1.86 − 3.23i)23-s + ⋯ |
| L(s) = 1 | + (−0.204 + 0.0909i)2-s + (−0.635 + 0.706i)4-s + (1.65 + 0.737i)5-s + (−1.26 − 0.267i)7-s + (0.134 − 0.414i)8-s − 0.405·10-s + (−0.985 − 0.166i)11-s + (−0.0400 + 0.381i)13-s + (0.281 − 0.0599i)14-s + (−0.0891 − 0.847i)16-s + (−1.26 + 0.916i)17-s + (−0.247 + 0.760i)19-s + (−1.57 + 0.700i)20-s + (0.216 − 0.0555i)22-s + (−0.389 − 0.675i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00433i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00433i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00115226 + 0.531559i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00115226 + 0.531559i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (3.27 + 0.553i)T \) |
| good | 2 | \( 1 + (0.288 - 0.128i)T + (1.33 - 1.48i)T^{2} \) |
| 5 | \( 1 + (-3.70 - 1.64i)T + (3.34 + 3.71i)T^{2} \) |
| 7 | \( 1 + (3.33 + 0.709i)T + (6.39 + 2.84i)T^{2} \) |
| 13 | \( 1 + (0.144 - 1.37i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (5.20 - 3.78i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.07 - 3.31i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (1.86 + 3.23i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.0729 - 0.0155i)T + (26.4 + 11.7i)T^{2} \) |
| 31 | \( 1 + (-0.390 + 3.71i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.381 + 1.17i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (9.11 - 1.93i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-0.228 + 0.395i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.46 + 1.63i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (-0.729 - 0.530i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (5.36 - 5.96i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (0.603 + 5.74i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (-2.92 - 5.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.66 + 5.56i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.34 - 13.3i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (10.0 - 4.47i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.143 - 1.36i)T + (-81.1 + 17.2i)T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + (7.53 - 3.35i)T + (64.9 - 72.0i)T^{2} \) |
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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
−10.14902832145782431836813511781, −9.884082252155495719460718097397, −8.976392102205863571291652830059, −8.157172991634048557322116524693, −6.88673982472043462244172891557, −6.41282177375271595741202485169, −5.49334389544747919111213988714, −4.15890571420937113982065556049, −3.08028494503808089961125791481, −2.14522994141274730155497868810,
0.25106705897357914654426306222, 1.87107594056148905012902826354, 2.87604010199922948996205297448, 4.71998437767867085207444612431, 5.29444794265520374157331139724, 6.08063363506701710749316199875, 6.86614908218732695497382061168, 8.457855380942469724906773910636, 9.127802581382164894047161295964, 9.694558936970992613104005823819
