| L(s) = 1 | − 4.05i·2-s − 0.408·4-s + 16.4·5-s + 47.3·7-s − 63.1i·8-s − 66.4i·10-s − 25.6i·11-s + 105. i·13-s − 191. i·14-s − 262.·16-s − 441.·17-s − 560.·19-s − 6.69·20-s − 103.·22-s − 764. i·23-s + ⋯ |
| L(s) = 1 | − 1.01i·2-s − 0.0255·4-s + 0.656·5-s + 0.965·7-s − 0.986i·8-s − 0.664i·10-s − 0.211i·11-s + 0.624i·13-s − 0.977i·14-s − 1.02·16-s − 1.52·17-s − 1.55·19-s − 0.0167·20-s − 0.214·22-s − 1.44i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.126229147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.126229147\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + (-3.44e3 + 496. i)T \) |
| good | 2 | \( 1 + 4.05iT - 16T^{2} \) |
| 5 | \( 1 - 16.4T + 625T^{2} \) |
| 7 | \( 1 - 47.3T + 2.40e3T^{2} \) |
| 11 | \( 1 + 25.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 105. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 441.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 560.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 764. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.38e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 950. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 632. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.02e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.95e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 4.32e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.11e3T + 7.89e6T^{2} \) |
| 61 | \( 1 - 4.27e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 867. iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 236.T + 2.54e7T^{2} \) |
| 73 | \( 1 + 6.21e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 8.91e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.08e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 6.26e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 7.89e3iT - 8.85e7T^{2} \) |
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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
−10.27987717790780156147761383533, −8.956095987910266032216993160803, −8.394234567017378421602425518980, −6.85112717665453787120586799072, −6.28396844639780520625878052327, −4.75772155443203869313026775457, −4.00234741100444117960276185874, −2.27142805049848203300396172725, −2.03057059674028894106696935191, −0.47853461608873808072279102465,
1.60441828950474929079218403447, 2.55568348196194814855375517988, 4.42311102748954414450688957069, 5.24460451316042251382848854470, 6.21302912317972973758881396428, 6.89887799806418384410250587949, 8.036648602296318123069952415617, 8.512802809627875085079447150785, 9.635233854625903742943375867933, 10.81610717002480686868197299517
