L(s) = 1 | + (−0.333 − 1.37i)2-s + (−1.77 + 0.916i)4-s + (1.20 + 0.498i)5-s + (2.59 + 2.59i)7-s + (1.85 + 2.13i)8-s + (0.283 − 1.82i)10-s + (−2.14 + 5.18i)11-s + (−0.984 + 0.407i)13-s + (2.69 − 4.43i)14-s + (2.31 − 3.25i)16-s − 0.979i·17-s + (5.68 − 2.35i)19-s + (−2.59 + 0.217i)20-s + (7.83 + 1.22i)22-s + (3.70 − 3.70i)23-s + ⋯ |
L(s) = 1 | + (−0.235 − 0.971i)2-s + (−0.888 + 0.458i)4-s + (0.538 + 0.223i)5-s + (0.980 + 0.980i)7-s + (0.655 + 0.755i)8-s + (0.0897 − 0.575i)10-s + (−0.647 + 1.56i)11-s + (−0.272 + 0.113i)13-s + (0.721 − 1.18i)14-s + (0.579 − 0.814i)16-s − 0.237i·17-s + (1.30 − 0.540i)19-s + (−0.580 + 0.0486i)20-s + (1.67 + 0.260i)22-s + (0.771 − 0.771i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20193 - 0.110770i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20193 - 0.110770i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.333 + 1.37i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.20 - 0.498i)T + (3.53 + 3.53i)T^{2} \) |
| 7 | \( 1 + (-2.59 - 2.59i)T + 7iT^{2} \) |
| 11 | \( 1 + (2.14 - 5.18i)T + (-7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (0.984 - 0.407i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + 0.979iT - 17T^{2} \) |
| 19 | \( 1 + (-5.68 + 2.35i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-3.70 + 3.70i)T - 23iT^{2} \) |
| 29 | \( 1 + (-1.17 - 2.83i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 - 1.54T + 31T^{2} \) |
| 37 | \( 1 + (8.23 + 3.41i)T + (26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-1.10 + 1.10i)T - 41iT^{2} \) |
| 43 | \( 1 + (3.47 - 8.37i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 - 3.15iT - 47T^{2} \) |
| 53 | \( 1 + (-2.55 + 6.16i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-8.95 - 3.70i)T + (41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (2.00 + 4.84i)T + (-43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (-1.14 - 2.76i)T + (-47.3 + 47.3i)T^{2} \) |
| 71 | \( 1 + (10.0 + 10.0i)T + 71iT^{2} \) |
| 73 | \( 1 + (-8.11 + 8.11i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.155iT - 79T^{2} \) |
| 83 | \( 1 + (5.13 - 2.12i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-6.15 - 6.15i)T + 89iT^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−11.85600590265987710263551867361, −10.84317197122316674498272735480, −9.924189734060448893351692006592, −9.204469477897133440438200958252, −8.158386913842760913847213975895, −7.12022615133385093240644930650, −5.28482563027680415054027546064, −4.70844028123895742707599765921, −2.77465513275185958205322515534, −1.86350126944493104921682849547,
1.14661228702317421506004904792, 3.63409321404641892189831502456, 5.13212309420609431652997933796, 5.69253153450401837490217681360, 7.12804255743706350898306407016, 7.936801482302265372396662862224, 8.725585700337955755975300701144, 9.906916209071632276880289487673, 10.66688513261908569358250862165, 11.70077590243138968932257409251