| L(s) = 1 | + (−6.86 + 21.1i)2-s + (42.7 − 31.0i)3-s + (14.7 + 10.6i)4-s + (597. + 1.84e3i)5-s + (362. + 1.11e3i)6-s + (−7.70e3 − 5.59e3i)7-s + (−9.53e3 + 6.92e3i)8-s + (−5.21e3 + 1.60e4i)9-s − 4.29e4·10-s + (4.72e4 − 1.10e4i)11-s + 960.·12-s + (−3.79e4 + 1.16e5i)13-s + (1.71e5 − 1.24e5i)14-s + (8.27e4 + 6.01e4i)15-s + (−7.80e4 − 2.40e5i)16-s + (2.54e4 + 7.82e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.303 + 0.934i)2-s + (0.304 − 0.221i)3-s + (0.0287 + 0.0208i)4-s + (0.427 + 1.31i)5-s + (0.114 + 0.351i)6-s + (−1.21 − 0.881i)7-s + (−0.822 + 0.597i)8-s + (−0.265 + 0.816i)9-s − 1.35·10-s + (0.973 − 0.227i)11-s + 0.0133·12-s + (−0.368 + 1.13i)13-s + (1.19 − 0.865i)14-s + (0.421 + 0.306i)15-s + (−0.297 − 0.916i)16-s + (0.0738 + 0.227i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 - 0.618i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.785 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.452408 + 1.30578i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.452408 + 1.30578i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-4.72e4 + 1.10e4i)T \) |
| good | 2 | \( 1 + (6.86 - 21.1i)T + (-414. - 300. i)T^{2} \) |
| 3 | \( 1 + (-42.7 + 31.0i)T + (6.08e3 - 1.87e4i)T^{2} \) |
| 5 | \( 1 + (-597. - 1.84e3i)T + (-1.58e6 + 1.14e6i)T^{2} \) |
| 7 | \( 1 + (7.70e3 + 5.59e3i)T + (1.24e7 + 3.83e7i)T^{2} \) |
| 13 | \( 1 + (3.79e4 - 1.16e5i)T + (-8.57e9 - 6.23e9i)T^{2} \) |
| 17 | \( 1 + (-2.54e4 - 7.82e4i)T + (-9.59e10 + 6.97e10i)T^{2} \) |
| 19 | \( 1 + (-6.77e5 + 4.92e5i)T + (9.97e10 - 3.06e11i)T^{2} \) |
| 23 | \( 1 - 1.33e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + (2.34e5 + 1.70e5i)T + (4.48e12 + 1.37e13i)T^{2} \) |
| 31 | \( 1 + (-3.54e5 + 1.09e6i)T + (-2.13e13 - 1.55e13i)T^{2} \) |
| 37 | \( 1 + (-1.44e7 - 1.05e7i)T + (4.01e13 + 1.23e14i)T^{2} \) |
| 41 | \( 1 + (-1.95e7 + 1.41e7i)T + (1.01e14 - 3.11e14i)T^{2} \) |
| 43 | \( 1 + 1.26e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (1.76e7 - 1.28e7i)T + (3.45e14 - 1.06e15i)T^{2} \) |
| 53 | \( 1 + (-9.12e6 + 2.80e7i)T + (-2.66e15 - 1.93e15i)T^{2} \) |
| 59 | \( 1 + (6.32e7 + 4.59e7i)T + (2.67e15 + 8.23e15i)T^{2} \) |
| 61 | \( 1 + (3.88e6 + 1.19e7i)T + (-9.46e15 + 6.87e15i)T^{2} \) |
| 67 | \( 1 + 1.92e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (-3.96e7 - 1.22e8i)T + (-3.70e16 + 2.69e16i)T^{2} \) |
| 73 | \( 1 + (-1.87e8 - 1.35e8i)T + (1.81e16 + 5.59e16i)T^{2} \) |
| 79 | \( 1 + (4.35e7 - 1.33e8i)T + (-9.69e16 - 7.04e16i)T^{2} \) |
| 83 | \( 1 + (1.12e8 + 3.45e8i)T + (-1.51e17 + 1.09e17i)T^{2} \) |
| 89 | \( 1 - 3.80e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (1.45e8 - 4.46e8i)T + (-6.15e17 - 4.46e17i)T^{2} \) |
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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
−18.82244048556707052376464955757, −17.19729492083369479588274090473, −16.29213097733270852303081931678, −14.60648710417280447039556295389, −13.64148482965212205072728599367, −11.25395056336920841183318522327, −9.449895892457495599502005570675, −7.26009652991081628975580134613, −6.50711893232212115906523478978, −2.93389513588370919640548613802,
0.906092528869018826835865871961, 3.10782895986431576202665849957, 5.91584862154424934597225619907, 9.152065247476187469699109471208, 9.655912487399489423147585853769, 12.03873787339145342871799961849, 12.77627332977874172879532131070, 15.06121499764223343600141444003, 16.40031362332458373191574602783, 18.05422950844587733123994084927
