| 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 | \( 1 + (-11.1 + 62.9i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.80761738471473379537600217888, −14.23213424166924098921387922894, −12.63887665470485342755307541118, −11.59626516548291808063499681409, −10.00374708619459914866121433884, −9.753128063155606639688239273749, −8.372464282198834270060882717426, −5.86101738981120090715707502114, −4.66887554076336065137421856876, −2.36590486683348285496550295967,
1.85497383287076393216452859612, 5.22279972350027477935399161181, 6.52310837163220674091954218943, 7.45346337405598955660737900625, 9.051052491769334937187299131772, 10.21591979810096149385330540051, 11.76164261165693013207419396042, 13.18901748325731551696080311004, 14.17417273490219947001399850895, 14.58077349817188532013408969542
