| L(s) = 1 | + (−21.3 + 65.8i)2-s + (373. − 271. i)3-s + (−2.22e3 − 1.61e3i)4-s + (2.90e3 + 8.92e3i)5-s + (9.87e3 + 3.03e4i)6-s + (1.51e4 + 1.10e4i)7-s + (3.91e4 − 2.84e4i)8-s + (1.10e4 − 3.39e4i)9-s − 6.49e5·10-s + (−4.74e5 + 2.44e5i)11-s − 1.26e6·12-s + (3.06e5 − 9.44e5i)13-s + (−1.04e6 + 7.61e5i)14-s + (3.50e6 + 2.54e6i)15-s + (−7.02e5 − 2.16e6i)16-s + (2.17e6 + 6.69e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.472 + 1.45i)2-s + (0.886 − 0.644i)3-s + (−1.08 − 0.788i)4-s + (0.415 + 1.27i)5-s + (0.518 + 1.59i)6-s + (0.340 + 0.247i)7-s + (0.422 − 0.306i)8-s + (0.0622 − 0.191i)9-s − 2.05·10-s + (−0.889 + 0.457i)11-s − 1.47·12-s + (0.229 − 0.705i)13-s + (−0.521 + 0.378i)14-s + (1.19 + 0.865i)15-s + (−0.167 − 0.515i)16-s + (0.371 + 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.335891 + 1.56970i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.335891 + 1.56970i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (4.74e5 - 2.44e5i)T \) |
| good | 2 | \( 1 + (21.3 - 65.8i)T + (-1.65e3 - 1.20e3i)T^{2} \) |
| 3 | \( 1 + (-373. + 271. i)T + (5.47e4 - 1.68e5i)T^{2} \) |
| 5 | \( 1 + (-2.90e3 - 8.92e3i)T + (-3.95e7 + 2.87e7i)T^{2} \) |
| 7 | \( 1 + (-1.51e4 - 1.10e4i)T + (6.11e8 + 1.88e9i)T^{2} \) |
| 13 | \( 1 + (-3.06e5 + 9.44e5i)T + (-1.44e12 - 1.05e12i)T^{2} \) |
| 17 | \( 1 + (-2.17e6 - 6.69e6i)T + (-2.77e13 + 2.01e13i)T^{2} \) |
| 19 | \( 1 + (1.25e7 - 9.11e6i)T + (3.59e13 - 1.10e14i)T^{2} \) |
| 23 | \( 1 - 3.26e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (-9.33e6 - 6.78e6i)T + (3.77e15 + 1.16e16i)T^{2} \) |
| 31 | \( 1 + (-8.88e7 + 2.73e8i)T + (-2.05e16 - 1.49e16i)T^{2} \) |
| 37 | \( 1 + (-2.61e8 - 1.90e8i)T + (5.49e16 + 1.69e17i)T^{2} \) |
| 41 | \( 1 + (1.62e8 - 1.17e8i)T + (1.70e17 - 5.23e17i)T^{2} \) |
| 43 | \( 1 - 1.25e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-1.92e9 + 1.40e9i)T + (7.63e17 - 2.35e18i)T^{2} \) |
| 53 | \( 1 + (-2.67e8 + 8.24e8i)T + (-7.49e18 - 5.44e18i)T^{2} \) |
| 59 | \( 1 + (2.04e9 + 1.48e9i)T + (9.31e18 + 2.86e19i)T^{2} \) |
| 61 | \( 1 + (-3.64e9 - 1.12e10i)T + (-3.52e19 + 2.55e19i)T^{2} \) |
| 67 | \( 1 + 4.95e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + (1.56e9 + 4.80e9i)T + (-1.86e20 + 1.35e20i)T^{2} \) |
| 73 | \( 1 + (6.23e9 + 4.53e9i)T + (9.69e19 + 2.98e20i)T^{2} \) |
| 79 | \( 1 + (1.00e9 - 3.08e9i)T + (-6.05e20 - 4.39e20i)T^{2} \) |
| 83 | \( 1 + (1.13e10 + 3.48e10i)T + (-1.04e21 + 7.56e20i)T^{2} \) |
| 89 | \( 1 + 2.31e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (-3.29e10 + 1.01e11i)T + (-5.78e21 - 4.20e21i)T^{2} \) |
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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
−18.34164455712463891957494915214, −17.12723986294860524410179591458, −15.10456477877023795876890748062, −14.68448594058439019637526666808, −13.20836394770846569779224291973, −10.40305134429438073577694786760, −8.379553360655124325122600298070, −7.44083896772215247295241911259, −5.96518191566153363737183336484, −2.47676268264753702073257671555,
0.887609821794632389650052049858, 2.71886843053335687869440268931, 4.56608706092049739620602993860, 8.660817531912053936044949329771, 9.356845568934774769457153710999, 10.88909838107024855262554921361, 12.58334477872454365199536116541, 13.85763188751599271298357204596, 15.85709032937951448634435796535, 17.45984277324993153518977727991
