| L(s) = 1 | + 2.05i·2-s + 11.7·4-s − 8.97i·5-s − 18.7·7-s + 57.0i·8-s + 18.4·10-s + 217. i·11-s − 276.·13-s − 38.4i·14-s + 71.4·16-s + 203. i·17-s + 27.7·19-s − 105. i·20-s − 446.·22-s − 9.99e2i·23-s + ⋯ |
| L(s) = 1 | + 0.513i·2-s + 0.736·4-s − 0.359i·5-s − 0.382·7-s + 0.891i·8-s + 0.184·10-s + 1.79i·11-s − 1.63·13-s − 0.196i·14-s + 0.278·16-s + 0.702i·17-s + 0.0767·19-s − 0.264i·20-s − 0.923·22-s − 1.88i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3407765651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3407765651\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 453. iT \) |
| good | 2 | \( 1 - 2.05iT - 16T^{2} \) |
| 5 | \( 1 + 8.97iT - 625T^{2} \) |
| 7 | \( 1 + 18.7T + 2.40e3T^{2} \) |
| 11 | \( 1 - 217. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 276.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 203. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 27.7T + 1.30e5T^{2} \) |
| 23 | \( 1 + 9.99e2iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 947. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 379.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 774.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 579. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.92e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 3.51e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.99e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 + 5.54e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 7.07e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 6.12e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.18e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.54e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.14e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 598. iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 6.26e3T + 8.85e7T^{2} \) |
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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
−10.46978908320000201401705150548, −10.04701322269120234642567793053, −8.926806793458269639794019270223, −7.85228625305577887600195666293, −7.08417929574782486001425899110, −6.46392309518105874563535810270, −5.16141155596436388050371519768, −4.42320599713932957562480602832, −2.67927189882429739705517974407, −1.85843251599860562973298530827,
0.07667292766943082499154877223, 1.48047988819809443084446433647, 3.01606029886827169121581027695, 3.24214982313024763777416558044, 5.04236961210312684888628934626, 6.07025231592575471260101951642, 7.03039705012436511432466458776, 7.70834704521663238549545826106, 9.071700281243208701826931904024, 9.834720764055987647045149297344
