| L(s) = 1 | + (−1.92 − 1.39i)2-s + (−0.729 − 2.24i)4-s + (10.2 + 4.41i)5-s + 25.3·7-s + (−7.60 + 23.3i)8-s + (−13.5 − 22.8i)10-s + (19.6 + 14.2i)11-s + (−67.4 + 49.0i)13-s + (−48.6 − 35.3i)14-s + (31.9 − 23.2i)16-s + (−12.7 + 39.3i)17-s + (−19.0 + 58.5i)19-s + (2.41 − 26.2i)20-s + (−17.8 − 54.8i)22-s + (36.6 + 26.6i)23-s + ⋯ |
| L(s) = 1 | + (−0.679 − 0.493i)2-s + (−0.0912 − 0.280i)4-s + (0.918 + 0.394i)5-s + 1.36·7-s + (−0.336 + 1.03i)8-s + (−0.429 − 0.721i)10-s + (0.538 + 0.391i)11-s + (−1.43 + 1.04i)13-s + (−0.929 − 0.675i)14-s + (0.499 − 0.363i)16-s + (−0.182 + 0.561i)17-s + (−0.229 + 0.706i)19-s + (0.0269 − 0.293i)20-s + (−0.172 − 0.531i)22-s + (0.332 + 0.241i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.40638 + 0.226737i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40638 + 0.226737i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-10.2 - 4.41i)T \) |
| good | 2 | \( 1 + (1.92 + 1.39i)T + (2.47 + 7.60i)T^{2} \) |
| 7 | \( 1 - 25.3T + 343T^{2} \) |
| 11 | \( 1 + (-19.6 - 14.2i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (67.4 - 49.0i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (12.7 - 39.3i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (19.0 - 58.5i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-36.6 - 26.6i)T + (3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (2.63 + 8.10i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (22.6 - 69.7i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-120. + 87.6i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (133. - 97.0i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 356.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-62.2 - 191. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (47.0 + 144. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-659. + 479. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (381. + 277. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-65.2 + 200. i)T + (-2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-31.0 - 95.5i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-944. - 686. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (225. + 694. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (255. - 785. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (870. + 632. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-58.4 - 179. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−11.51708427738861605505985814012, −10.82297345728295934416674968956, −9.830390503721692529181577765921, −9.228181436416633270911556136279, −8.095515286254453394456215708344, −6.84271284009857497198093696823, −5.53056623651886920422672279583, −4.53363392343534782113804944422, −2.24573871199560462731904781371, −1.53554748847173894376580455181,
0.790552138239371322251476288212, 2.55598080515321752025197097848, 4.52084528892738648660141524602, 5.51143830371877088331677566026, 6.93911765606649240770547207258, 7.87022299611111122180302801804, 8.760229319886870353925661360848, 9.525891084128213272227818571157, 10.56841085648835828846575279140, 11.78361307961667258822092505512
