| L(s) = 1 | + (11.8 + 36.3i)2-s + (17.5 + 12.7i)3-s + (−770. + 559. i)4-s + (−276. + 849. i)5-s + (−256. + 788. i)6-s + (42.1 − 30.6i)7-s + (−1.36e4 − 9.90e3i)8-s + (−5.93e3 − 1.82e4i)9-s − 3.41e4·10-s + (2.67e4 + 4.05e4i)11-s − 2.06e4·12-s + (3.64e4 + 1.12e5i)13-s + (1.61e3 + 1.17e3i)14-s + (−1.56e4 + 1.13e4i)15-s + (4.85e4 − 1.49e5i)16-s + (4.02e4 − 1.23e5i)17-s + ⋯ |
| L(s) = 1 | + (0.522 + 1.60i)2-s + (0.124 + 0.0907i)3-s + (−1.50 + 1.09i)4-s + (−0.197 + 0.607i)5-s + (−0.0806 + 0.248i)6-s + (0.00663 − 0.00482i)7-s + (−1.17 − 0.854i)8-s + (−0.301 − 0.928i)9-s − 1.08·10-s + (0.551 + 0.834i)11-s − 0.287·12-s + (0.354 + 1.09i)13-s + (0.0112 + 0.00815i)14-s + (−0.0798 + 0.0580i)15-s + (0.185 − 0.570i)16-s + (0.116 − 0.359i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0899i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0819712 + 1.81889i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0819712 + 1.81889i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-2.67e4 - 4.05e4i)T \) |
| good | 2 | \( 1 + (-11.8 - 36.3i)T + (-414. + 300. i)T^{2} \) |
| 3 | \( 1 + (-17.5 - 12.7i)T + (6.08e3 + 1.87e4i)T^{2} \) |
| 5 | \( 1 + (276. - 849. i)T + (-1.58e6 - 1.14e6i)T^{2} \) |
| 7 | \( 1 + (-42.1 + 30.6i)T + (1.24e7 - 3.83e7i)T^{2} \) |
| 13 | \( 1 + (-3.64e4 - 1.12e5i)T + (-8.57e9 + 6.23e9i)T^{2} \) |
| 17 | \( 1 + (-4.02e4 + 1.23e5i)T + (-9.59e10 - 6.97e10i)T^{2} \) |
| 19 | \( 1 + (-6.63e5 - 4.82e5i)T + (9.97e10 + 3.06e11i)T^{2} \) |
| 23 | \( 1 + 9.51e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + (-3.81e6 + 2.76e6i)T + (4.48e12 - 1.37e13i)T^{2} \) |
| 31 | \( 1 + (2.42e6 + 7.46e6i)T + (-2.13e13 + 1.55e13i)T^{2} \) |
| 37 | \( 1 + (-9.82e6 + 7.14e6i)T + (4.01e13 - 1.23e14i)T^{2} \) |
| 41 | \( 1 + (2.77e7 + 2.01e7i)T + (1.01e14 + 3.11e14i)T^{2} \) |
| 43 | \( 1 - 2.51e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-1.72e7 - 1.25e7i)T + (3.45e14 + 1.06e15i)T^{2} \) |
| 53 | \( 1 + (-1.19e7 - 3.66e7i)T + (-2.66e15 + 1.93e15i)T^{2} \) |
| 59 | \( 1 + (-1.78e7 + 1.29e7i)T + (2.67e15 - 8.23e15i)T^{2} \) |
| 61 | \( 1 + (-3.47e6 + 1.07e7i)T + (-9.46e15 - 6.87e15i)T^{2} \) |
| 67 | \( 1 + 9.99e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-8.21e6 + 2.52e7i)T + (-3.70e16 - 2.69e16i)T^{2} \) |
| 73 | \( 1 + (2.80e7 - 2.03e7i)T + (1.81e16 - 5.59e16i)T^{2} \) |
| 79 | \( 1 + (9.24e7 + 2.84e8i)T + (-9.69e16 + 7.04e16i)T^{2} \) |
| 83 | \( 1 + (-1.90e7 + 5.87e7i)T + (-1.51e17 - 1.09e17i)T^{2} \) |
| 89 | \( 1 - 4.98e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (-3.32e8 - 1.02e9i)T + (-6.15e17 + 4.46e17i)T^{2} \) |
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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
−18.51421631336458902986428541821, −17.29129875413316560447227761268, −15.95404076152127594215600318168, −14.77807218541558535332940082432, −13.97781656819306663631676214687, −11.96675313511373439291036990283, −9.298123551785282255457364054841, −7.42960205472340437735221563399, −6.16235087242489239094160098640, −4.01897792168478277838746339921,
1.02916571589501635738477221650, 3.13036615764111784472119629896, 5.08317135291485232631254038773, 8.558859122185780604707007196844, 10.43347029471584186933411837661, 11.69480927402361444147805186461, 13.03396481540933536684140905456, 14.07539549163124576706487212436, 16.26821397390992335691209520525, 18.19539634712127198100401062128
