| L(s) = 1 | + (−0.707 − 1.22i)2-s + (1.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s + (7.24 − 4.18i)5-s − 2.44i·6-s + (−6.74 − 1.88i)7-s + 2.82·8-s + (1.5 + 2.59i)9-s + (−10.2 − 5.91i)10-s + (−3 + 5.19i)11-s + (−2.99 + 1.73i)12-s + 17.8i·13-s + (2.46 + 9.58i)14-s + 14.4·15-s + (−2.00 − 3.46i)16-s + (−16.2 − 9.37i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.5 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (1.44 − 0.836i)5-s − 0.408i·6-s + (−0.963 − 0.268i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−1.02 − 0.591i)10-s + (−0.272 + 0.472i)11-s + (−0.249 + 0.144i)12-s + 1.37i·13-s + (0.176 + 0.684i)14-s + 0.965·15-s + (−0.125 − 0.216i)16-s + (−0.955 − 0.551i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.784 + 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.05547 - 0.366754i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05547 - 0.366754i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 + 1.22i)T \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 7 | \( 1 + (6.74 + 1.88i)T \) |
| good | 5 | \( 1 + (-7.24 + 4.18i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (3 - 5.19i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 17.8iT - 169T^{2} \) |
| 17 | \( 1 + (16.2 + 9.37i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (14.7 - 8.51i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (6.72 + 11.6i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 33.9T + 841T^{2} \) |
| 31 | \( 1 + (-12.7 - 7.37i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (2.98 + 5.17i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 35.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 15.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (28.7 - 16.6i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-17.2 + 29.9i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-23.6 - 13.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (34.9 - 20.1i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-57.1 + 99.0i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 18.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + (101. + 58.5i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-44.1 - 76.5i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 75.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (18 - 10.3i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 30.5iT - 9.40e3T^{2} \) |
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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
−16.00950219902075500525944375486, −14.09477525083185604248842534886, −13.33688960435497473853652846671, −12.33830774036683535277034811095, −10.41022451160148627894738851518, −9.530279792776025170513439817410, −8.768268876148122137929799949480, −6.56597622883056463986598583795, −4.51522153840729080074153825979, −2.20604359483068962765831611063,
2.72031556201279849766172301524, 5.85989696359579339620966549555, 6.71536788738059700592190572363, 8.453261571995771634331487690457, 9.743878526286321490894530839445, 10.59955010809873457371118535736, 12.99939544595648790195484888839, 13.58812681151318234801985838313, 14.85053438489955024356384319604, 15.75441535323854672418462197524
