L(s) = 1 | + (−0.115 + 0.356i)2-s + (0.695 + 0.505i)4-s + (−0.393 − 1.21i)5-s + (−0.563 + 0.409i)8-s + (0.309 − 0.951i)9-s + 0.477·10-s + (−0.187 − 0.982i)11-s + (0.450 − 1.38i)13-s + (0.184 + 0.569i)16-s + (0.303 + 0.220i)18-s + (−1.56 + 1.13i)19-s + (0.338 − 1.04i)20-s + (0.371 + 0.0469i)22-s + 1.93·23-s + (−0.505 + 0.367i)25-s + (0.442 + 0.321i)26-s + ⋯ |
L(s) = 1 | + (−0.115 + 0.356i)2-s + (0.695 + 0.505i)4-s + (−0.393 − 1.21i)5-s + (−0.563 + 0.409i)8-s + (0.309 − 0.951i)9-s + 0.477·10-s + (−0.187 − 0.982i)11-s + (0.450 − 1.38i)13-s + (0.184 + 0.569i)16-s + (0.303 + 0.220i)18-s + (−1.56 + 1.13i)19-s + (0.338 − 1.04i)20-s + (0.371 + 0.0469i)22-s + 1.93·23-s + (−0.505 + 0.367i)25-s + (0.442 + 0.321i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.014078670\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.014078670\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.187 + 0.982i)T \) |
| 79 | \( 1 + (-0.309 + 0.951i)T \) |
good | 2 | \( 1 + (0.115 - 0.356i)T + (-0.809 - 0.587i)T^{2} \) |
| 3 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (0.393 + 1.21i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.450 + 1.38i)T + (-0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (1.56 - 1.13i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 - 1.93T + T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.613 - 1.88i)T + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 - 0.125T + T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-1.50 - 1.09i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + 0.851T + T^{2} \) |
| 97 | \( 1 + (-0.331 + 1.01i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−10.51093014150006132857870614151, −9.026456553954469796358144830968, −8.556608637949895889920796865397, −7.978443370659723474111494026631, −6.88384502289417794830297230295, −6.02429196586892523821345388082, −5.16666649573776746886430860571, −3.81763044149622388577638137818, −3.04990361758688374312513655076, −1.15179391303695001800305845858,
1.98258010952793066961948010652, 2.60527746224242096665925153334, 4.01567302380562497444522141501, 5.03099235916851967993349007389, 6.47725464079317745532104205442, 6.91421293125189772042793580231, 7.56274519182694084835807218583, 8.936255710551546332362228435487, 9.761841725756523863113519743693, 10.76456863361160350925629690455