| L(s) = 1 | + (−19.5 − 11.3i)2-s + (255. + 443. i)4-s + (4.17e3 − 2.40e3i)5-s + (−335. + 580. i)7-s − 1.15e4i·8-s − 1.09e5·10-s + (2.02e5 + 1.16e5i)11-s + (−1.53e5 − 2.66e5i)13-s + (1.31e4 − 7.58e3i)14-s + (−1.31e5 + 2.27e5i)16-s − 6.72e5i·17-s − 1.55e6·19-s + (2.13e6 + 1.23e6i)20-s + (−2.63e6 − 4.57e6i)22-s + (−4.82e6 + 2.78e6i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (1.33 − 0.770i)5-s + (−0.0199 + 0.0345i)7-s − 0.353i·8-s − 1.09·10-s + (1.25 + 0.724i)11-s + (−0.414 − 0.717i)13-s + (0.0244 − 0.0141i)14-s + (−0.125 + 0.216i)16-s − 0.473i·17-s − 0.626·19-s + (0.667 + 0.385i)20-s + (−0.512 − 0.886i)22-s + (−0.750 + 0.433i)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
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.216793793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.216793793\) |
| \(L(6)\) |
|
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 | \( 1 + (-4.17e3 + 2.40e3i)T + (4.88e6 - 8.45e6i)T^{2} \) |
| 7 | \( 1 + (335. - 580. i)T + (-1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 + (-2.02e5 - 1.16e5i)T + (1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + (1.53e5 + 2.66e5i)T + (-6.89e10 + 1.19e11i)T^{2} \) |
| 17 | \( 1 + 6.72e5iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 1.55e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + (4.82e6 - 2.78e6i)T + (2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 + (2.57e7 + 1.48e7i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (1.54e7 + 2.67e7i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + 8.56e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + (3.10e7 - 1.79e7i)T + (6.71e15 - 1.16e16i)T^{2} \) |
| 43 | \( 1 + (-1.83e7 + 3.17e7i)T + (-1.08e16 - 1.87e16i)T^{2} \) |
| 47 | \( 1 + (-2.84e7 - 1.64e7i)T + (2.62e16 + 4.55e16i)T^{2} \) |
| 53 | \( 1 + 4.59e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-4.23e8 + 2.44e8i)T + (2.55e17 - 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-3.06e7 + 5.30e7i)T + (-3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (-3.35e8 - 5.80e8i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 + 1.23e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 1.08e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + (-9.33e8 + 1.61e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (9.48e8 + 5.47e8i)T + (7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 + 5.19e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (-5.37e9 + 9.30e9i)T + (-3.68e19 - 6.38e19i)T^{2} \) |
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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
−10.17680008882065861066849327970, −9.545184479567050328329111056187, −8.876100602689127579347428917998, −7.56199891210200113718379022285, −6.32199326784568535522705425038, −5.27348408014613244752135247046, −3.92063589939593618978670806833, −2.22797816407952610130335995620, −1.55775362319940372850606041070, −0.29212001342111436997964555689,
1.40798938771966769554566240707, 2.22196201559395416659309581281, 3.74460414106230244264868685997, 5.48988066755380045122887426930, 6.41761797693352113873449698005, 7.01639255529082352002462575310, 8.642793555024805574097013229142, 9.350085312247388409924784418493, 10.31527089820579290141844826720, 11.07599418502475536326143923929
