| L(s) = 1 | + (−19.5 + 11.3i)2-s + (−156. + 185. i)3-s + (255. − 443. i)4-s + (1.21e3 + 698. i)5-s + (968. − 5.41e3i)6-s + (−1.06e4 − 1.83e4i)7-s + 1.15e4i·8-s + (−9.96e3 − 5.82e4i)9-s − 3.16e4·10-s + (−2.32e5 + 1.34e5i)11-s + (4.22e4 + 1.17e5i)12-s + (−2.66e5 + 4.61e5i)13-s + (4.15e5 + 2.40e5i)14-s + (−3.19e5 + 1.15e5i)15-s + (−1.31e5 − 2.27e5i)16-s − 1.68e6i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.644 + 0.764i)3-s + (0.249 − 0.433i)4-s + (0.387 + 0.223i)5-s + (0.124 − 0.696i)6-s + (−0.631 − 1.09i)7-s + 0.353i·8-s + (−0.168 − 0.985i)9-s − 0.316·10-s + (−1.44 + 0.832i)11-s + (0.169 + 0.470i)12-s + (−0.717 + 1.24i)13-s + (0.773 + 0.446i)14-s + (−0.420 + 0.151i)15-s + (−0.125 − 0.216i)16-s − 1.18i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.762i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.646 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.485009 + 0.224689i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.485009 + 0.224689i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (19.5 - 11.3i)T \) |
| 3 | \( 1 + (156. - 185. i)T \) |
| 5 | \( 1 + (-1.21e3 - 698. i)T \) |
| good | 7 | \( 1 + (1.06e4 + 1.83e4i)T + (-1.41e8 + 2.44e8i)T^{2} \) |
| 11 | \( 1 + (2.32e5 - 1.34e5i)T + (1.29e10 - 2.24e10i)T^{2} \) |
| 13 | \( 1 + (2.66e5 - 4.61e5i)T + (-6.89e10 - 1.19e11i)T^{2} \) |
| 17 | \( 1 + 1.68e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 3.14e5T + 6.13e12T^{2} \) |
| 23 | \( 1 + (7.09e6 + 4.09e6i)T + (2.07e13 + 3.58e13i)T^{2} \) |
| 29 | \( 1 + (3.12e7 - 1.80e7i)T + (2.10e14 - 3.64e14i)T^{2} \) |
| 31 | \( 1 + (-2.27e7 + 3.94e7i)T + (-4.09e14 - 7.09e14i)T^{2} \) |
| 37 | \( 1 - 2.77e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + (1.24e8 + 7.20e7i)T + (6.71e15 + 1.16e16i)T^{2} \) |
| 43 | \( 1 + (-7.18e7 - 1.24e8i)T + (-1.08e16 + 1.87e16i)T^{2} \) |
| 47 | \( 1 + (-2.95e8 + 1.70e8i)T + (2.62e16 - 4.55e16i)T^{2} \) |
| 53 | \( 1 - 1.14e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-4.76e7 - 2.74e7i)T + (2.55e17 + 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-2.78e8 - 4.82e8i)T + (-3.56e17 + 6.17e17i)T^{2} \) |
| 67 | \( 1 + (-8.59e8 + 1.48e9i)T + (-9.11e17 - 1.57e18i)T^{2} \) |
| 71 | \( 1 - 2.29e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 2.42e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + (-1.81e9 - 3.14e9i)T + (-4.73e18 + 8.19e18i)T^{2} \) |
| 83 | \( 1 + (-4.51e9 + 2.60e9i)T + (7.75e18 - 1.34e19i)T^{2} \) |
| 89 | \( 1 - 6.74e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (1.24e8 + 2.14e8i)T + (-3.68e19 + 6.38e19i)T^{2} \) |
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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
−11.97148379600491837858007845641, −10.75399941207470270466625064025, −9.953369707465515681110136315421, −9.414505886601345902056141662774, −7.50311307123278224314107006948, −6.71185520825431386065704055688, −5.33038745497429192991998190887, −4.18996550177407380328452404359, −2.37873116980862912986944740668, −0.43222024552390645723499263768,
0.42283965769147897089932477144, 1.98949459679133269417074487484, 2.96839384656824872293671674995, 5.44567134606688714416752504339, 6.00873972868060089634059976596, 7.69349007956779403645986359918, 8.460827401499254462732358259519, 9.947713258719726555653487914581, 10.75999905791247277994583556367, 12.06885249096244560737076361817
