| L(s) = 1 | + (−19.5 + 11.3i)2-s + (255. − 443. i)4-s + (−3.14e3 − 1.81e3i)5-s + (1.16e4 + 2.01e4i)7-s + 1.15e4i·8-s + 8.21e4·10-s + (−5.40e4 + 3.12e4i)11-s + (8.50e4 − 1.47e5i)13-s + (−4.55e5 − 2.62e5i)14-s + (−1.31e5 − 2.27e5i)16-s + 2.66e6i·17-s + 7.66e5·19-s + (−1.60e6 + 9.29e5i)20-s + (7.06e5 − 1.22e6i)22-s + (1.21e6 + 7.01e5i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−1.00 − 0.580i)5-s + (0.690 + 1.19i)7-s + 0.353i·8-s + 0.821·10-s + (−0.335 + 0.193i)11-s + (0.229 − 0.396i)13-s + (−0.846 − 0.488i)14-s + (−0.125 − 0.216i)16-s + 1.87i·17-s + 0.309·19-s + (−0.503 + 0.290i)20-s + (0.137 − 0.237i)22-s + (0.188 + 0.109i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.6980685746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6980685746\) |
| \(L(6)\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + (3.14e3 + 1.81e3i)T + (4.88e6 + 8.45e6i)T^{2} \) |
| 7 | \( 1 + (-1.16e4 - 2.01e4i)T + (-1.41e8 + 2.44e8i)T^{2} \) |
| 11 | \( 1 + (5.40e4 - 3.12e4i)T + (1.29e10 - 2.24e10i)T^{2} \) |
| 13 | \( 1 + (-8.50e4 + 1.47e5i)T + (-6.89e10 - 1.19e11i)T^{2} \) |
| 17 | \( 1 - 2.66e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 7.66e5T + 6.13e12T^{2} \) |
| 23 | \( 1 + (-1.21e6 - 7.01e5i)T + (2.07e13 + 3.58e13i)T^{2} \) |
| 29 | \( 1 + (4.18e6 - 2.41e6i)T + (2.10e14 - 3.64e14i)T^{2} \) |
| 31 | \( 1 + (-2.09e7 + 3.62e7i)T + (-4.09e14 - 7.09e14i)T^{2} \) |
| 37 | \( 1 - 5.01e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + (1.29e8 + 7.46e7i)T + (6.71e15 + 1.16e16i)T^{2} \) |
| 43 | \( 1 + (-9.93e7 - 1.72e8i)T + (-1.08e16 + 1.87e16i)T^{2} \) |
| 47 | \( 1 + (-1.34e8 + 7.75e7i)T + (2.62e16 - 4.55e16i)T^{2} \) |
| 53 | \( 1 + 4.21e7iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-2.52e8 - 1.46e8i)T + (2.55e17 + 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-2.65e8 - 4.59e8i)T + (-3.56e17 + 6.17e17i)T^{2} \) |
| 67 | \( 1 + (2.61e8 - 4.52e8i)T + (-9.11e17 - 1.57e18i)T^{2} \) |
| 71 | \( 1 + 5.71e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 2.18e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + (9.82e8 + 1.70e9i)T + (-4.73e18 + 8.19e18i)T^{2} \) |
| 83 | \( 1 + (-1.89e9 + 1.09e9i)T + (7.75e18 - 1.34e19i)T^{2} \) |
| 89 | \( 1 - 2.38e8iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (-4.42e9 - 7.65e9i)T + (-3.68e19 + 6.38e19i)T^{2} \) |
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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
−11.45896265541903724329801333521, −10.40587850432846907717884854667, −9.089474703794478390574517519309, −8.224099760287627029456029213705, −7.81819926355228436726870163082, −6.16901622693011987192488672882, −5.21022525980398197985655415805, −3.98542955959836740623521217694, −2.33009787946758678035535238538, −1.10250795689897022950008502002,
0.23057522378590587347054331222, 1.14678649981146485865098246756, 2.76519609759874665612831206697, 3.80775187517767246760239190755, 4.89982121507683219343118015071, 6.91540698339935396896583432780, 7.44964751470711091515826344941, 8.359082406823076839344402350935, 9.649380919388830199802163181321, 10.74895943499443352903495539081
