L(s) = 1 | + (0.128 + 0.0457i)2-s + (−0.979 − 0.203i)3-s + (−0.761 − 0.619i)4-s + (−0.116 − 0.0709i)6-s + (−0.140 − 0.231i)8-s + (0.917 + 0.398i)9-s + (0.619 + 0.761i)12-s + (0.192 + 0.924i)16-s + (−1.02 − 0.530i)17-s + (0.0997 + 0.0931i)18-s + (−0.105 + 1.54i)19-s + (0.631 − 0.224i)23-s + (0.0905 + 0.254i)24-s + (−0.816 − 0.576i)27-s + (0.187 + 0.900i)31-s + (−0.0543 + 0.395i)32-s + ⋯ |
L(s) = 1 | + (0.128 + 0.0457i)2-s + (−0.979 − 0.203i)3-s + (−0.761 − 0.619i)4-s + (−0.116 − 0.0709i)6-s + (−0.140 − 0.231i)8-s + (0.917 + 0.398i)9-s + (0.619 + 0.761i)12-s + (0.192 + 0.924i)16-s + (−1.02 − 0.530i)17-s + (0.0997 + 0.0931i)18-s + (−0.105 + 1.54i)19-s + (0.631 − 0.224i)23-s + (0.0905 + 0.254i)24-s + (−0.816 − 0.576i)27-s + (0.187 + 0.900i)31-s + (−0.0543 + 0.395i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6709050302\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6709050302\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.979 + 0.203i)T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + (-0.269 - 0.962i)T \) |
good | 2 | \( 1 + (-0.128 - 0.0457i)T + (0.775 + 0.631i)T^{2} \) |
| 7 | \( 1 + (-0.334 + 0.942i)T^{2} \) |
| 11 | \( 1 + (0.576 - 0.816i)T^{2} \) |
| 13 | \( 1 + (-0.0682 + 0.997i)T^{2} \) |
| 17 | \( 1 + (1.02 + 0.530i)T + (0.576 + 0.816i)T^{2} \) |
| 19 | \( 1 + (0.105 - 1.54i)T + (-0.990 - 0.136i)T^{2} \) |
| 23 | \( 1 + (-0.631 + 0.224i)T + (0.775 - 0.631i)T^{2} \) |
| 29 | \( 1 + (0.0682 + 0.997i)T^{2} \) |
| 31 | \( 1 + (-0.187 - 0.900i)T + (-0.917 + 0.398i)T^{2} \) |
| 37 | \( 1 + (0.854 + 0.519i)T^{2} \) |
| 41 | \( 1 + (-0.460 - 0.887i)T^{2} \) |
| 43 | \( 1 + (0.203 + 0.979i)T^{2} \) |
| 53 | \( 1 + (0.953 - 1.56i)T + (-0.460 - 0.887i)T^{2} \) |
| 59 | \( 1 + (-0.203 + 0.979i)T^{2} \) |
| 61 | \( 1 + (-1.31 + 0.368i)T + (0.854 - 0.519i)T^{2} \) |
| 67 | \( 1 + (-0.334 - 0.942i)T^{2} \) |
| 71 | \( 1 + (0.775 - 0.631i)T^{2} \) |
| 73 | \( 1 + (0.682 - 0.730i)T^{2} \) |
| 79 | \( 1 + (-0.911 - 0.125i)T + (0.962 + 0.269i)T^{2} \) |
| 83 | \( 1 + (-1.51 + 0.786i)T + (0.576 - 0.816i)T^{2} \) |
| 89 | \( 1 + (0.990 - 0.136i)T^{2} \) |
| 97 | \( 1 + (-0.917 - 0.398i)T^{2} \) |
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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
−8.874612762194027535553397907512, −8.039586073588168722725483493538, −7.12679180085977398485740850966, −6.36419677601640150975802744915, −5.81307151037071886084227396446, −4.97226634484269845882999584659, −4.48475159373169474623126298320, −3.52943647899004667431413361028, −2.01620680239087578237365035330, −0.946492287936220952679393078044,
0.58236829194796138436614684910, 2.24725643457268324350634371059, 3.42680268544471480868415410660, 4.28245004041285622653999019829, 4.82339148777278987819527740114, 5.53374911143709631793214722205, 6.53029456616572988991707353141, 7.08356215280073830405599849013, 7.991166269069583118703527706294, 8.850504191939963897613017466445