| L(s) = 1 | + (5.75 − 7.21i)2-s + (8.62 + 10.8i)3-s + (−11.8 − 51.7i)4-s + (90.5 + 43.6i)5-s + 127.·6-s − 71.2·7-s + (−175. − 84.3i)8-s + (11.4 − 50.3i)9-s + (835. − 402. i)10-s + (−24.3 + 106. i)11-s + (457. − 574. i)12-s + (−693. − 333. i)13-s + (−409. + 513. i)14-s + (309. + 1.35e3i)15-s + (−85.3 + 41.1i)16-s + (−1.30e3 + 630. i)17-s + ⋯ |
| L(s) = 1 | + (1.01 − 1.27i)2-s + (0.553 + 0.693i)3-s + (−0.369 − 1.61i)4-s + (1.62 + 0.780i)5-s + 1.44·6-s − 0.549·7-s + (−0.967 − 0.465i)8-s + (0.0472 − 0.207i)9-s + (2.64 − 1.27i)10-s + (−0.0606 + 0.265i)11-s + (0.917 − 1.15i)12-s + (−1.13 − 0.547i)13-s + (−0.558 + 0.700i)14-s + (0.355 + 1.55i)15-s + (−0.0833 + 0.0401i)16-s + (−1.09 + 0.529i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.13487 - 1.60801i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.13487 - 1.60801i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (5.50e3 + 1.08e4i)T \) |
| good | 2 | \( 1 + (-5.75 + 7.21i)T + (-7.12 - 31.1i)T^{2} \) |
| 3 | \( 1 + (-8.62 - 10.8i)T + (-54.0 + 236. i)T^{2} \) |
| 5 | \( 1 + (-90.5 - 43.6i)T + (1.94e3 + 2.44e3i)T^{2} \) |
| 7 | \( 1 + 71.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + (24.3 - 106. i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (693. + 333. i)T + (2.31e5 + 2.90e5i)T^{2} \) |
| 17 | \( 1 + (1.30e3 - 630. i)T + (8.85e5 - 1.11e6i)T^{2} \) |
| 19 | \( 1 + (314. + 1.37e3i)T + (-2.23e6 + 1.07e6i)T^{2} \) |
| 23 | \( 1 + (236. - 1.03e3i)T + (-5.79e6 - 2.79e6i)T^{2} \) |
| 29 | \( 1 + (2.65e3 - 3.33e3i)T + (-4.56e6 - 1.99e7i)T^{2} \) |
| 31 | \( 1 + (7.83 - 9.82i)T + (-6.37e6 - 2.79e7i)T^{2} \) |
| 37 | \( 1 - 1.12e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (1.25e4 - 1.57e4i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 47 | \( 1 + (-2.11e3 - 9.28e3i)T + (-2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (2.90e3 - 1.39e3i)T + (2.60e8 - 3.26e8i)T^{2} \) |
| 59 | \( 1 + (-3.85e4 + 1.85e4i)T + (4.45e8 - 5.58e8i)T^{2} \) |
| 61 | \( 1 + (2.22e4 + 2.79e4i)T + (-1.87e8 + 8.23e8i)T^{2} \) |
| 67 | \( 1 + (9.93e3 + 4.35e4i)T + (-1.21e9 + 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-5.05e3 - 2.21e4i)T + (-1.62e9 + 7.82e8i)T^{2} \) |
| 73 | \( 1 + (-4.98e4 - 2.39e4i)T + (1.29e9 + 1.62e9i)T^{2} \) |
| 79 | \( 1 - 4.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (1.08e4 + 1.35e4i)T + (-8.76e8 + 3.84e9i)T^{2} \) |
| 89 | \( 1 + (1.02e4 + 1.28e4i)T + (-1.24e9 + 5.44e9i)T^{2} \) |
| 97 | \( 1 + (-3.76e3 + 1.65e4i)T + (-7.73e9 - 3.72e9i)T^{2} \) |
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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
−14.56202854759305574821785973499, −13.44232365361383973953858825682, −12.73578517250922506589015144046, −11.00871173922667967998948317871, −9.987383491599811069046692645472, −9.457705326059119910097895627261, −6.49057723784695086997284663512, −4.93293041305909833315186136964, −3.21258829552421052311391722244, −2.20523488866171288225215746093,
2.21258643756903817241166594601, 4.75531955892296580183846594458, 5.99722233964472265917832369510, 7.07617441051862577615855861829, 8.549062383886137962382973756091, 9.858167028108855162970807130521, 12.47565804504808848285003185596, 13.32385519063639097473235509744, 13.77388300473306797754648915793, 14.77314664628742504153725740769
