| L(s) = 1 | + (−50.6 + 130. i)3-s − 1.33e3·5-s − 5.39e3i·7-s + (−1.45e4 − 1.32e4i)9-s + 7.37e4i·11-s + 1.62e5i·13-s + (6.74e4 − 1.74e5i)15-s + 5.57e4i·17-s + 1.59e5·19-s + (7.05e5 + 2.72e5i)21-s + 1.18e6·23-s − 1.77e5·25-s + (2.46e6 − 1.23e6i)27-s + 2.26e6·29-s + 5.08e6i·31-s + ⋯ |
| L(s) = 1 | + (−0.360 + 0.932i)3-s − 0.953·5-s − 0.848i·7-s + (−0.739 − 0.672i)9-s + 1.51i·11-s + 1.57i·13-s + (0.343 − 0.889i)15-s + 0.161i·17-s + 0.279·19-s + (0.791 + 0.306i)21-s + 0.880·23-s − 0.0908·25-s + (0.894 − 0.447i)27-s + 0.595·29-s + 0.988i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.5252216919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5252216919\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (50.6 - 130. i)T \) |
| good | 5 | \( 1 + 1.33e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.39e3iT - 4.03e7T^{2} \) |
| 11 | \( 1 - 7.37e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.62e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 5.57e4iT - 1.18e11T^{2} \) |
| 19 | \( 1 - 1.59e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.18e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.26e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.08e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 - 1.75e7iT - 1.29e14T^{2} \) |
| 41 | \( 1 - 2.76e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 3.45e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 7.35e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.69e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.26e8iT - 8.66e15T^{2} \) |
| 61 | \( 1 + 1.22e8iT - 1.16e16T^{2} \) |
| 67 | \( 1 + 2.00e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 6.75e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.22e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.22e7iT - 1.19e17T^{2} \) |
| 83 | \( 1 + 5.99e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 8.82e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 - 9.09e8T + 7.60e17T^{2} \) |
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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
−11.62079505258252253438922517228, −10.45608962603735727808665798048, −9.726249315162680110107367953990, −8.659785912828912991072036168968, −7.33992046266720856235742137872, −6.58056105540788380556477363012, −4.69655692836428750950469975316, −4.40799234807748917459416727281, −3.23409514702229059598968006592, −1.39975301045123854567125100079,
0.16627367851175998491905249137, 0.833785530387092367244806733816, 2.54471344198761665487869540216, 3.48501607406147423853632332525, 5.33124850460552636872112730276, 5.93708632429396368510450018410, 7.31228187359130952620785915037, 8.150396092714894050413713052228, 8.832473019584768379859172894349, 10.59539889563607049925971243551
