| L(s) = 1 | + (−49.0 − 35.6i)2-s + (−8.11 + 24.9i)3-s + (503. + 1.54e3i)4-s + (1.03e4 − 7.54e3i)5-s + (1.28e3 − 936. i)6-s + (−1.35e4 − 4.16e4i)7-s + (−7.87e3 + 2.42e4i)8-s + (1.42e5 + 1.03e5i)9-s − 7.78e5·10-s + (−4.81e5 − 2.31e5i)11-s − 4.27e4·12-s + (−1.50e6 − 1.09e6i)13-s + (−8.20e5 + 2.52e6i)14-s + (1.04e5 + 3.20e5i)15-s + (3.94e6 − 2.86e6i)16-s + (2.03e6 − 1.47e6i)17-s + ⋯ |
| L(s) = 1 | + (−1.08 − 0.787i)2-s + (−0.0192 + 0.0593i)3-s + (0.245 + 0.755i)4-s + (1.48 − 1.07i)5-s + (0.0676 − 0.0491i)6-s + (−0.304 − 0.936i)7-s + (−0.0849 + 0.261i)8-s + (0.805 + 0.585i)9-s − 2.46·10-s + (−0.901 − 0.432i)11-s − 0.0496·12-s + (−1.12 − 0.815i)13-s + (−0.407 + 1.25i)14-s + (0.0354 + 0.109i)15-s + (0.940 − 0.683i)16-s + (0.347 − 0.252i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.0898461 - 0.875275i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0898461 - 0.875275i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (4.81e5 + 2.31e5i)T \) |
| good | 2 | \( 1 + (49.0 + 35.6i)T + (632. + 1.94e3i)T^{2} \) |
| 3 | \( 1 + (8.11 - 24.9i)T + (-1.43e5 - 1.04e5i)T^{2} \) |
| 5 | \( 1 + (-1.03e4 + 7.54e3i)T + (1.50e7 - 4.64e7i)T^{2} \) |
| 7 | \( 1 + (1.35e4 + 4.16e4i)T + (-1.59e9 + 1.16e9i)T^{2} \) |
| 13 | \( 1 + (1.50e6 + 1.09e6i)T + (5.53e11 + 1.70e12i)T^{2} \) |
| 17 | \( 1 + (-2.03e6 + 1.47e6i)T + (1.05e13 - 3.25e13i)T^{2} \) |
| 19 | \( 1 + (1.92e6 - 5.93e6i)T + (-9.42e13 - 6.84e13i)T^{2} \) |
| 23 | \( 1 + 4.31e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (2.93e6 + 9.04e6i)T + (-9.87e15 + 7.17e15i)T^{2} \) |
| 31 | \( 1 + (-6.20e6 - 4.51e6i)T + (7.85e15 + 2.41e16i)T^{2} \) |
| 37 | \( 1 + (-1.25e8 - 3.85e8i)T + (-1.43e17 + 1.04e17i)T^{2} \) |
| 41 | \( 1 + (-4.61e7 + 1.42e8i)T + (-4.45e17 - 3.23e17i)T^{2} \) |
| 43 | \( 1 + 5.75e7T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-6.27e8 + 1.93e9i)T + (-2.00e18 - 1.45e18i)T^{2} \) |
| 53 | \( 1 + (1.20e9 + 8.78e8i)T + (2.86e18 + 8.81e18i)T^{2} \) |
| 59 | \( 1 + (-3.01e8 - 9.27e8i)T + (-2.43e19 + 1.77e19i)T^{2} \) |
| 61 | \( 1 + (-7.48e9 + 5.43e9i)T + (1.34e19 - 4.13e19i)T^{2} \) |
| 67 | \( 1 - 1.64e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + (-2.27e8 + 1.65e8i)T + (7.14e19 - 2.19e20i)T^{2} \) |
| 73 | \( 1 + (5.17e9 + 1.59e10i)T + (-2.53e20 + 1.84e20i)T^{2} \) |
| 79 | \( 1 + (-1.85e10 - 1.34e10i)T + (2.31e20 + 7.11e20i)T^{2} \) |
| 83 | \( 1 + (2.24e10 - 1.63e10i)T + (3.97e20 - 1.22e21i)T^{2} \) |
| 89 | \( 1 + 7.28e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-8.69e10 - 6.31e10i)T + (2.21e21 + 6.80e21i)T^{2} \) |
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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
−17.30796763717019916089268948629, −16.43078901528242853945267087768, −13.79181802717521332952019327318, −12.63351437266366188940071420912, −10.20448434317466131429037851714, −9.934701690879444957828485740285, −8.040149224194397909817623992397, −5.27945753369376854241317703805, −2.08817015903929504785083363000, −0.61371707532949906826523471507,
2.24255577826212542510806094550, 6.03975559487460395303355929043, 7.21212485207502119822680409233, 9.408138929961592105919138944496, 10.09381983645327801509861554199, 12.68350751801381248279872476670, 14.56849539634184938765797040637, 15.75441188062139903751702472569, 17.42340661598915553517218944146, 18.20382353647592556903722917370
