| L(s) = 1 | + (−19.5 − 11.3i)2-s + (16.6 + 242. i)3-s + (255. + 443. i)4-s + (2.72e3 − 1.57e3i)5-s + (2.41e3 − 4.93e3i)6-s + (−1.76e3 + 3.06e3i)7-s − 1.15e4i·8-s + (−5.84e4 + 8.09e3i)9-s − 7.11e4·10-s + (1.01e5 + 5.88e4i)11-s + (−1.03e5 + 6.94e4i)12-s + (1.40e5 + 2.42e5i)13-s + (6.93e4 − 4.00e4i)14-s + (4.26e5 + 6.33e5i)15-s + (−1.31e5 + 2.27e5i)16-s + 2.62e6i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.0687 + 0.997i)3-s + (0.249 + 0.433i)4-s + (0.871 − 0.503i)5-s + (0.310 − 0.635i)6-s + (−0.105 + 0.182i)7-s − 0.353i·8-s + (−0.990 + 0.137i)9-s − 0.711·10-s + (0.632 + 0.365i)11-s + (−0.414 + 0.279i)12-s + (0.377 + 0.653i)13-s + (0.128 − 0.0744i)14-s + (0.561 + 0.834i)15-s + (−0.125 + 0.216i)16-s + 1.84i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0370 - 0.999i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.0370 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.954562 + 0.919869i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.954562 + 0.919869i\) |
| \(L(6)\) |
|
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 | \( 1 + (-16.6 - 242. i)T \) |
| good | 5 | \( 1 + (-2.72e3 + 1.57e3i)T + (4.88e6 - 8.45e6i)T^{2} \) |
| 7 | \( 1 + (1.76e3 - 3.06e3i)T + (-1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 + (-1.01e5 - 5.88e4i)T + (1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + (-1.40e5 - 2.42e5i)T + (-6.89e10 + 1.19e11i)T^{2} \) |
| 17 | \( 1 - 2.62e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 1.16e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + (6.93e6 - 4.00e6i)T + (2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 + (-2.58e7 - 1.49e7i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (9.71e6 + 1.68e7i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + 2.58e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + (3.60e7 - 2.08e7i)T + (6.71e15 - 1.16e16i)T^{2} \) |
| 43 | \( 1 + (-9.70e7 + 1.68e8i)T + (-1.08e16 - 1.87e16i)T^{2} \) |
| 47 | \( 1 + (-2.63e8 - 1.51e8i)T + (2.62e16 + 4.55e16i)T^{2} \) |
| 53 | \( 1 - 2.93e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-4.61e8 + 2.66e8i)T + (2.55e17 - 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-5.88e8 + 1.01e9i)T + (-3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (9.91e8 + 1.71e9i)T + (-9.11e17 + 1.57e18i)T^{2} \) |
| 71 | \( 1 + 1.40e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 6.62e7T + 4.29e18T^{2} \) |
| 79 | \( 1 + (2.94e9 - 5.10e9i)T + (-4.73e18 - 8.19e18i)T^{2} \) |
| 83 | \( 1 + (-2.24e9 - 1.29e9i)T + (7.75e18 + 1.34e19i)T^{2} \) |
| 89 | \( 1 + 8.23e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (6.95e9 - 1.20e10i)T + (-3.68e19 - 6.38e19i)T^{2} \) |
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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
−16.86235015692985114212574641458, −15.51491895844238825380269160146, −14.01867532573532465416364750890, −12.32912265472672438475422371242, −10.71227080164419263965403979208, −9.563308408843295656298843247272, −8.586957907161055876073032619010, −6.01361702542906465648343580157, −4.00959150960637542568399952362, −1.80547518043024740498930227835,
0.73058395315191136907555815427, 2.49873817544010628789556060786, 5.92849115647731593224658871335, 7.02754528840577214198758964916, 8.616345229672201391400761136210, 10.18551708470563750436416741073, 11.79204850084749136116601318789, 13.57437599697071137585020666442, 14.37420545099211937379605280278, 16.22729565618462854158235594064
