| L(s) = 1 | + (0.721 − 0.262i)2-s + (2.90 − 0.749i)3-s + (−2.61 + 2.19i)4-s + (2.96 + 4.02i)5-s + (1.89 − 1.30i)6-s + (3.17 − 3.78i)7-s + (−2.84 + 4.92i)8-s + (7.87 − 4.35i)9-s + (3.19 + 2.12i)10-s + (7.37 + 1.30i)11-s + (−5.94 + 8.32i)12-s + (−6.15 + 16.9i)13-s + (1.29 − 3.56i)14-s + (11.6 + 9.45i)15-s + (1.61 − 9.14i)16-s + (−10.2 − 17.8i)17-s + ⋯ |
| L(s) = 1 | + (0.360 − 0.131i)2-s + (0.968 − 0.249i)3-s + (−0.653 + 0.548i)4-s + (0.593 + 0.804i)5-s + (0.316 − 0.217i)6-s + (0.453 − 0.540i)7-s + (−0.355 + 0.615i)8-s + (0.875 − 0.483i)9-s + (0.319 + 0.212i)10-s + (0.670 + 0.118i)11-s + (−0.495 + 0.693i)12-s + (−0.473 + 1.30i)13-s + (0.0926 − 0.254i)14-s + (0.776 + 0.630i)15-s + (0.100 − 0.571i)16-s + (−0.605 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.284i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.16760 + 0.314366i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.16760 + 0.314366i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.90 + 0.749i)T \) |
| 5 | \( 1 + (-2.96 - 4.02i)T \) |
| good | 2 | \( 1 + (-0.721 + 0.262i)T + (3.06 - 2.57i)T^{2} \) |
| 7 | \( 1 + (-3.17 + 3.78i)T + (-8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (-7.37 - 1.30i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (6.15 - 16.9i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (10.2 + 17.8i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-13.4 + 23.3i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (10.0 - 8.46i)T + (91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (13.1 + 36.2i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (34.2 - 28.7i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (39.3 - 22.7i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-18.6 + 51.2i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-27.9 - 4.92i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (21.0 + 17.6i)T + (383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 44.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + (15.0 - 2.66i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (44.3 + 37.1i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (7.91 - 21.7i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-91.6 + 52.9i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-67.6 - 39.0i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-64.5 + 23.5i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (71.0 - 25.8i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-95.3 - 55.0i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-86.5 - 15.2i)T + (8.84e3 + 3.21e3i)T^{2} \) |
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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
−13.52251807701265773007057254878, −12.14148975219088979593088029963, −11.17419003617386850807925418226, −9.489210014857644679500745004254, −9.130619307331257114779920233057, −7.54388827153662881335152233830, −6.83264012999688668416573766074, −4.77800615802369272727657766064, −3.59783684349552891269919761026, −2.20522862821481304791695721777,
1.67938800030837697473177107561, 3.74629417135705233597642390357, 5.02356544733016463591766126491, 5.95389035026908476603531666819, 7.950727663441275158071936286037, 8.853604767168181935539947226531, 9.610579391218881043261122985934, 10.56397505657788426025246023897, 12.51228599035610721710835995160, 12.96361964562287047455894084872
