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
−11.07599418502475536326143923929, −10.31527089820579290141844826720, −9.350085312247388409924784418493, −8.642793555024805574097013229142, −7.01639255529082352002462575310, −6.41761797693352113873449698005, −5.48988066755380045122887426930, −3.74460414106230244264868685997, −2.22196201559395416659309581281, −1.40798938771966769554566240707,
0.29212001342111436997964555689, 1.55775362319940372850606041070, 2.22797816407952610130335995620, 3.92063589939593618978670806833, 5.27348408014613244752135247046, 6.32199326784568535522705425038, 7.56199891210200113718379022285, 8.876100602689127579347428917998, 9.545184479567050328329111056187, 10.17680008882065861066849327970