| L(s) = 1 | + (−0.590 − 0.429i)2-s + (2.78 − 8.55i)3-s + (−9.72 − 29.9i)4-s + (−77.5 + 56.3i)5-s + (−5.31 + 3.86i)6-s + (29.5 + 91.0i)7-s + (−14.3 + 44.0i)8-s + (−65.5 − 47.6i)9-s + 69.9·10-s + (−386. + 106. i)11-s − 283.·12-s + (−573. − 416. i)13-s + (21.5 − 66.4i)14-s + (266. + 820. i)15-s + (−787. + 571. i)16-s + (815. − 592. i)17-s + ⋯ |
| L(s) = 1 | + (−0.104 − 0.0758i)2-s + (0.178 − 0.549i)3-s + (−0.303 − 0.935i)4-s + (−1.38 + 1.00i)5-s + (−0.0602 + 0.0438i)6-s + (0.228 + 0.702i)7-s + (−0.0791 + 0.243i)8-s + (−0.269 − 0.195i)9-s + 0.221·10-s + (−0.964 + 0.264i)11-s − 0.567·12-s + (−0.940 − 0.683i)13-s + (0.0294 − 0.0906i)14-s + (0.305 + 0.941i)15-s + (−0.768 + 0.558i)16-s + (0.684 − 0.497i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0174257 + 0.102631i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0174257 + 0.102631i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.78 + 8.55i)T \) |
| 11 | \( 1 + (386. - 106. i)T \) |
| good | 2 | \( 1 + (0.590 + 0.429i)T + (9.88 + 30.4i)T^{2} \) |
| 5 | \( 1 + (77.5 - 56.3i)T + (965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-29.5 - 91.0i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (573. + 416. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-815. + 592. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-580. + 1.78e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + 2.81e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-1.56e3 - 4.80e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-1.58e3 - 1.15e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (649. + 1.99e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-1.40e3 + 4.33e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 2.24e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.39e3 - 7.38e3i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.34e4 + 9.78e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-605. - 1.86e3i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (3.25e4 - 2.36e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 - 5.50e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-2.16e4 + 1.57e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-2.03e4 - 6.26e4i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (7.55e3 + 5.49e3i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (3.13e4 - 2.27e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + 1.37e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-4.52e3 - 3.29e3i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−15.07242636167751527076303631053, −14.09385202531975930268793012745, −12.38439330241812691494844784963, −11.28169241638426622580728464422, −10.04179108508712285773293841140, −8.225257109416943595350856188342, −7.07016380323749001396838937732, −5.16347346102177772965298003696, −2.75109752626870555623635045288, −0.06207925930904903018015924759,
3.69570443467343069702597426923, 4.71146373460754586735187142523, 7.77750039305281511209323925772, 8.199510819506230932370088600897, 9.886034581496366612171044173109, 11.67277950800595582537042633803, 12.50176713465661668990263787967, 13.90939575225949635667047215188, 15.49088266537125611485799134860, 16.49161862281927973020286844358
