L(s) = 1 | − 3i·3-s − 10.4·5-s + (9.40 + 15.9i)7-s − 9·9-s + 51.5·11-s − 46.4·13-s + 31.2i·15-s − 99.1i·17-s − 12.5i·19-s + (47.8 − 28.2i)21-s + 129. i·23-s − 16.2·25-s + 27i·27-s − 207. i·29-s − 139.·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.932·5-s + (0.507 + 0.861i)7-s − 0.333·9-s + 1.41·11-s − 0.990·13-s + 0.538i·15-s − 1.41i·17-s − 0.151i·19-s + (0.497 − 0.293i)21-s + 1.17i·23-s − 0.130·25-s + 0.192i·27-s − 1.32i·29-s − 0.808·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.662636235\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.662636235\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 7 | \( 1 + (-9.40 - 15.9i)T \) |
good | 5 | \( 1 + 10.4T + 125T^{2} \) |
| 11 | \( 1 - 51.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 99.1iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 12.5iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 129. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 207. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 152. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 317. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 366.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 103.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 24.2iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 339. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 25.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 660.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 0.771iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 342. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 951. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 244. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.17e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.40e3iT - 9.12e5T^{2} \) |
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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
−9.312331064894470539661880465378, −8.236583515816428056169260107023, −7.59046709825695552378671271764, −6.94062305297469510633223737493, −5.92575855589641684558319700812, −4.98615746841326132189281402103, −4.07955927884145669651464738503, −2.94858587016658079023358143454, −1.91336462354786876585050095583, −0.65865902876984768459880577933,
0.63501723153105311203966907869, 1.95123593854085578252024491722, 3.59059353377230735846612655498, 4.03515925859026621090179637021, 4.75743608239851788983363190678, 5.94968293230395985976543695402, 7.01563383067532402298363173249, 7.59081448630206875107741692072, 8.567872313338973359401826246928, 9.140826538560558661891936614592