| L(s) = 1 | + (−0.483 + 1.32i)2-s + (−0.391 − 2.97i)3-s + (−1.53 − 1.28i)4-s + (7.52 + 1.32i)5-s + (4.14 + 0.918i)6-s + (4.40 − 3.69i)7-s + (2.44 − 1.41i)8-s + (−8.69 + 2.33i)9-s + (−5.40 + 9.35i)10-s + (8.02 − 1.41i)11-s + (−3.22 + 5.06i)12-s + (−22.0 + 8.02i)13-s + (2.78 + 7.64i)14-s + (0.998 − 22.8i)15-s + (0.694 + 3.93i)16-s + (6.39 + 3.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.241 + 0.664i)2-s + (−0.130 − 0.991i)3-s + (−0.383 − 0.321i)4-s + (1.50 + 0.265i)5-s + (0.690 + 0.153i)6-s + (0.629 − 0.528i)7-s + (0.306 − 0.176i)8-s + (−0.965 + 0.258i)9-s + (−0.540 + 0.935i)10-s + (0.729 − 0.128i)11-s + (−0.268 + 0.421i)12-s + (−1.69 + 0.617i)13-s + (0.198 + 0.546i)14-s + (0.0665 − 1.52i)15-s + (0.0434 + 0.246i)16-s + (0.376 + 0.217i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0725i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.15734 - 0.0420098i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15734 - 0.0420098i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.483 - 1.32i)T \) |
| 3 | \( 1 + (0.391 + 2.97i)T \) |
| good | 5 | \( 1 + (-7.52 - 1.32i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-4.40 + 3.69i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (-8.02 + 1.41i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (22.0 - 8.02i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-6.39 - 3.69i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (7.80 + 13.5i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (19.9 - 23.8i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (9.68 - 26.6i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-12.2 - 10.2i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-5.99 + 10.3i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-2.82 - 7.76i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (7.82 + 44.3i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (43.4 + 51.7i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 16.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-57.3 - 10.1i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-54.6 + 45.8i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (47.1 - 17.1i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-35.4 - 20.4i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (49.7 + 86.2i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (72.4 + 26.3i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-36.0 + 99.1i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (0.302 - 0.174i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-11.1 - 62.9i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−14.58077349817188532013408969542, −14.17417273490219947001399850895, −13.18901748325731551696080311004, −11.76164261165693013207419396042, −10.21591979810096149385330540051, −9.051052491769334937187299131772, −7.45346337405598955660737900625, −6.52310837163220674091954218943, −5.22279972350027477935399161181, −1.85497383287076393216452859612,
2.36590486683348285496550295967, 4.66887554076336065137421856876, 5.86101738981120090715707502114, 8.372464282198834270060882717426, 9.753128063155606639688239273749, 10.00374708619459914866121433884, 11.59626516548291808063499681409, 12.63887665470485342755307541118, 14.23213424166924098921387922894, 14.80761738471473379537600217888
