| L(s) = 1 | + (0.483 + 1.32i)2-s + (2.82 + 1.01i)3-s + (−1.53 + 1.28i)4-s + (0.891 − 0.157i)5-s + (0.0210 + 4.24i)6-s + (−5.50 − 4.61i)7-s + (−2.44 − 1.41i)8-s + (6.95 + 5.71i)9-s + (0.639 + 1.10i)10-s + (7.28 + 1.28i)11-s + (−5.62 + 2.08i)12-s + (−18.3 − 6.68i)13-s + (3.47 − 9.54i)14-s + (2.67 + 0.458i)15-s + (0.694 − 3.93i)16-s + (12.7 − 7.37i)17-s + ⋯ |
| L(s) = 1 | + (0.241 + 0.664i)2-s + (0.941 + 0.337i)3-s + (−0.383 + 0.321i)4-s + (0.178 − 0.0314i)5-s + (0.00350 + 0.707i)6-s + (−0.785 − 0.659i)7-s + (−0.306 − 0.176i)8-s + (0.772 + 0.635i)9-s + (0.0639 + 0.110i)10-s + (0.662 + 0.116i)11-s + (−0.468 + 0.173i)12-s + (−1.41 − 0.514i)13-s + (0.248 − 0.681i)14-s + (0.178 + 0.0305i)15-s + (0.0434 − 0.246i)16-s + (0.751 − 0.434i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.34736 + 0.739465i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34736 + 0.739465i\) |
| \(L(2)\) |
|
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 | \( 1 + (-2.82 - 1.01i)T \) |
| good | 5 | \( 1 + (-0.891 + 0.157i)T + (23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (5.50 + 4.61i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-7.28 - 1.28i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (18.3 + 6.68i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-12.7 + 7.37i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-6.58 + 11.4i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (15.0 + 17.9i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-13.5 - 37.1i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (9.20 - 7.72i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-33.0 - 57.1i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (26.1 - 71.8i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-9.66 + 54.8i)T + (-1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (0.204 - 0.243i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 0.264iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (62.3 - 10.9i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (42.4 + 35.6i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-58.6 - 21.3i)T + (3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-57.0 + 32.9i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-22.7 + 39.3i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-22.8 + 8.33i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (28.1 + 77.2i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-115. - 66.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (11.9 - 67.8i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−15.14230233585643474511354836823, −14.28045955692318643373865487986, −13.39930193688468393081427389171, −12.25632153193234486054849359414, −10.12656897887176628796372688112, −9.406400328376012984056517854884, −7.87338221131156393807998679724, −6.80178964552566361997204718415, −4.81357597313444444844623199547, −3.21802727489498168280848481960,
2.25542782338801403643698443051, 3.83156481441931742622711093152, 6.02849691942766652989607639362, 7.72629482969175581643907410224, 9.355026620339179140761724945212, 9.852126168685976212721726087904, 11.93413972558226761810750516540, 12.54084534890505198039683490799, 13.85693470653973845162807717293, 14.58002428003038643008766201163
