| L(s) = 1 | + (−0.618 − 1.90i)2-s + (−0.396 − 0.287i)3-s + (−3.23 + 2.35i)4-s + (−1.54 + 4.75i)5-s + (−0.302 + 0.931i)6-s + (19.1 − 13.9i)7-s + (6.47 + 4.70i)8-s + (−8.26 − 25.4i)9-s + 10.0·10-s + (−35.9 − 6.11i)11-s + 1.95·12-s + (−22.0 − 67.9i)13-s + (−38.3 − 27.8i)14-s + (1.98 − 1.43i)15-s + (4.94 − 15.2i)16-s + (−8.55 + 26.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.0762 − 0.0554i)3-s + (−0.404 + 0.293i)4-s + (−0.138 + 0.425i)5-s + (−0.0206 + 0.0634i)6-s + (1.03 − 0.751i)7-s + (0.286 + 0.207i)8-s + (−0.306 − 0.942i)9-s + 0.316·10-s + (−0.985 − 0.167i)11-s + 0.0471·12-s + (−0.470 − 1.44i)13-s + (−0.731 − 0.531i)14-s + (0.0341 − 0.0247i)15-s + (0.0772 − 0.237i)16-s + (−0.122 + 0.375i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.362357 - 0.950530i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.362357 - 0.950530i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.618 + 1.90i)T \) |
| 5 | \( 1 + (1.54 - 4.75i)T \) |
| 11 | \( 1 + (35.9 + 6.11i)T \) |
| good | 3 | \( 1 + (0.396 + 0.287i)T + (8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 + (-19.1 + 13.9i)T + (105. - 326. i)T^{2} \) |
| 13 | \( 1 + (22.0 + 67.9i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (8.55 - 26.3i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-3.30 - 2.39i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + 74.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-229. + 166. i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (92.0 + 283. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (89.3 - 64.9i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-151. - 110. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 131.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-97.8 - 71.0i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-165. - 509. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-421. + 305. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (216. - 665. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 - 710.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-62.2 + 191. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (458. - 332. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (349. + 1.07e3i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (444. - 1.36e3i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 - 575.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (255. + 785. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−12.64165922808025300088317993018, −11.59387261217012300432387729109, −10.67755385066721554271095149769, −9.929608732158537628102839128171, −8.272864725868494467391013401880, −7.57849576713423574970466174922, −5.77313484074140804684990930249, −4.21624195450614836228712778887, −2.71175328366351753105260510423, −0.60299120426095129596649922322,
2.07568525085981082178266969251, 4.74203248781539095572739966582, 5.33351224649979611190821190633, 7.07685724392759830260844392764, 8.214515329238273234440850507935, 8.924264269346698547870547359014, 10.37017991438765200337213060560, 11.49937097666952748167497350257, 12.52053127123546729443856330192, 13.95059152564324384444796271436
