| L(s) = 1 | + (−19.5 + 11.3i)2-s + (208. − 124. i)3-s + (255. − 443. i)4-s + (−4.47e3 − 2.58e3i)5-s + (−2.68e3 + 4.79e3i)6-s + (7.93e3 + 1.37e4i)7-s + 1.15e4i·8-s + (2.81e4 − 5.19e4i)9-s + 1.16e5·10-s + (−1.21e5 + 6.99e4i)11-s + (−1.70e3 − 1.24e5i)12-s + (−3.43e5 + 5.95e5i)13-s + (−3.11e5 − 1.79e5i)14-s + (−1.25e6 + 1.71e4i)15-s + (−1.31e5 − 2.27e5i)16-s − 9.11e5i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.859 − 0.511i)3-s + (0.249 − 0.433i)4-s + (−1.43 − 0.826i)5-s + (−0.345 + 0.617i)6-s + (0.472 + 0.818i)7-s + 0.353i·8-s + (0.476 − 0.879i)9-s + 1.16·10-s + (−0.752 + 0.434i)11-s + (−0.00683 − 0.499i)12-s + (−0.925 + 1.60i)13-s + (−0.578 − 0.334i)14-s + (−1.65 + 0.0225i)15-s + (−0.125 − 0.216i)16-s − 0.642i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.0162099 + 0.0999009i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0162099 + 0.0999009i\) |
| \(L(6)\) |
|
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 | \( 1 + (-208. + 124. i)T \) |
| good | 5 | \( 1 + (4.47e3 + 2.58e3i)T + (4.88e6 + 8.45e6i)T^{2} \) |
| 7 | \( 1 + (-7.93e3 - 1.37e4i)T + (-1.41e8 + 2.44e8i)T^{2} \) |
| 11 | \( 1 + (1.21e5 - 6.99e4i)T + (1.29e10 - 2.24e10i)T^{2} \) |
| 13 | \( 1 + (3.43e5 - 5.95e5i)T + (-6.89e10 - 1.19e11i)T^{2} \) |
| 17 | \( 1 + 9.11e5iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 3.96e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + (2.79e5 + 1.61e5i)T + (2.07e13 + 3.58e13i)T^{2} \) |
| 29 | \( 1 + (1.96e7 - 1.13e7i)T + (2.10e14 - 3.64e14i)T^{2} \) |
| 31 | \( 1 + (4.80e5 - 8.32e5i)T + (-4.09e14 - 7.09e14i)T^{2} \) |
| 37 | \( 1 - 8.29e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + (4.29e7 + 2.47e7i)T + (6.71e15 + 1.16e16i)T^{2} \) |
| 43 | \( 1 + (1.64e7 + 2.85e7i)T + (-1.08e16 + 1.87e16i)T^{2} \) |
| 47 | \( 1 + (8.48e6 - 4.89e6i)T + (2.62e16 - 4.55e16i)T^{2} \) |
| 53 | \( 1 + 1.50e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (6.48e8 + 3.74e8i)T + (2.55e17 + 4.42e17i)T^{2} \) |
| 61 | \( 1 + (3.97e7 + 6.88e7i)T + (-3.56e17 + 6.17e17i)T^{2} \) |
| 67 | \( 1 + (1.31e8 - 2.28e8i)T + (-9.11e17 - 1.57e18i)T^{2} \) |
| 71 | \( 1 - 1.74e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 1.37e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + (2.41e9 + 4.18e9i)T + (-4.73e18 + 8.19e18i)T^{2} \) |
| 83 | \( 1 + (4.80e9 - 2.77e9i)T + (7.75e18 - 1.34e19i)T^{2} \) |
| 89 | \( 1 - 3.34e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (6.16e9 + 1.06e10i)T + (-3.68e19 + 6.38e19i)T^{2} \) |
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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
−16.73786222115780304334020839846, −15.47678417101076356204985783358, −14.64767162795425803994097411050, −12.66246827985630516099475062730, −11.62493674682044567297494143098, −9.221910827232684382047585960141, −8.280438764903187959201137586095, −7.18757938313609848096400104292, −4.54531880389023683113858412803, −2.06088104474018727677836216144,
0.04710630738326260869514210760, 2.79877334379376025231418722940, 4.12086723678550276284513452917, 7.61068481017722376140944089740, 8.140138999694224419837418880051, 10.34037284498689335669540411951, 10.97968718795100662245284974077, 12.91390390203576059796545398342, 14.79873966431005221501318211970, 15.44282588199116987716328553446
